Yeah, it really doesn't work that way....

The Mueller Report, despite being written by Angry Democrats and Trump Haters, and with unlimited money behind it ($35,000,000), didn’t lay a glove on me. I DID NOTHING WRONG. If the partisan Dems ever tried to Impeach, I would first head to the U.S. Supreme Court. Not only......

— Donald J. Trump (@realDonaldTrump) April 24, 2019

So, here's a good general analysis of the Constitutional provisions for impeachment of a sitting President; although actually, the thrust of the analysis is on removal, not impeachment.

First:  grounds for impeachment.

... the operative legal standard to apply to an impeachment of a sitting President is "treason, bribery, or other high crimes and misdemeanors." There is substantial difference of opinion over the interpretation of these words.

The interpretations run broadly into four categories.   First is the idea that Congress alone can decide what a "high crime and misdemeanor" is.  Legal scholars don't like this, because it means the President serves at the pleasure of Congress.  I have to point out, though, the history of Bush v. Gore, where the Supreme Court entered into a political issue (who won the count in Florida?) and put their thumb on the scales.  They clearly thought this was a one-off opinion (they declared it to be so), and just as clearly thought they were doing everybody a favor.  The public reaction to that opinion led the Court to not do something so bold or paternalistic again.  There is a history of the Court finding out it has gotten too far away from public opinion.  Reversing a removal of a President from office because the Court's majority doesn't like the way the House framed the Articles of Impeachment would probably be a bridge too far for many observers, and represent a Constitutional crisis on its own (and call into question the legitimacy of the Judiciary).  So while the analysis is dismissive of this interpretation because legal scholars find it discomfiting, I wouldn't discard it quite so readily.

The second category is that the President must have committed an indictable crime (a point made, obliquely, by Kellyanne Conway this morning; this White House is scared to death of the prospect of impeachment).  But the word "misdemeanor," say others, means an indictable crime is not required.  This argument centers on the debates over the language of this article.  "Corruption" was rejected, as was "maladministration," and finally "misdemeanors" was approved, although that was not a term of criminal law at the time.

The fourth category is that the offense must relate to official Presidential duties.

All of these, of course, go solely to the question of impeachment.  But the issue becomes reviewable (at least the analysis seems to agree), when (and only when) the trial in the Senate results in removal.

There are a few subtle issues, here.  First, it seems to be agreed that judicial review is not available for impeachment, only for removal from office following impeachment.  However, the rules by which the House and Senate conduct the removal process (impeachment and trial, respectively), are well within the power of each house to establish.  That seems pretty clear from the language of Art. 1, Sec. 3:

"The Senate shall have the sole Power to try all Impeachments..."

I also want to add here that:

The House of Representatives shall chuse their Speaker and other Officers;and shall have the sole Power of Impeachment.

Art. 1, Sec. 2, U.S. Constitution.  Which seems to pretty much give the House the power to impeach on whatever grounds they choose, and in whatever manner they decide upon (just as the House alone can pick its "Speaker and other Officers.")  Can the Court disagree with the House's definition of "high crimes and misdemeanors"?  That would certainly interfere with the "sole power", wouldn't it?  (The analysis of possible judicial review based on Nixon, below, goes only to the obligation of the Senate to "try all Impeachments," rather than take an extraordinary step to vote for removal without any trial.  This indicates the Court would be very reluctant to interfere with any process that fits the broad outlines of Art. 1.)

That's often read as preventing judicial review of such a trial; but it clearly limits any possible judicial review to outcome, not process.  If the Senate has sole power to try impeachments, it has sole power to determine how to try impeachments, and the Courts have no final authority to review that process; only to, as below, be sure some process (v. no process) is followed.

Which makes the question of judicial review of an impeachment trial even thornier.  As I said, the Court has found itself ahead of (or behind) public opinion before, and is usually careful to circumscribe its role in political fights, especially.  The Court may side with the Administration in the Census question fight; whether it will enter into a fight between Congress and the President over how to build the border wall, is another matter (the Court certainly won't decide how Congress appropriates money, or not, for such issues).  Pay attention to the language here, dicta in the case before the Court, but part of the paucity of case law on Presidential impeachments (since no President has ever been removed from office):

"Finally, as applied to the special case of the President, the majority's argument merely points out that, were the Senate to convict the President without any kind of trial, a Constitutional crisis might well result. It hardly follows that the Court ought to refrain from upholding the Constitution in all impeachment cases. Nor does it follow that, in cases of Presidential impeachment, the Justices ought to abandon their constitutional responsibilities because the Senate has precipitated a crisis."

The argument there posits an extreme case:  removal from Presidential office without a trial.  The Senate may have sole power to try impeachments, but the argument would be the Senate must have a trial.  Being a political act, an impeachment trial would have to be perceived as fair and in accordance with at least rough notions of due process.  But could the Court review the trial procedure for compliance with Federal Rules of Procedure?  Or could it merely step into a Constitutional crisis caused by slighting the Constitution as much as possible, as in disposing of the effort of a Senate trial altogether?  The argument there turns on the Senate precipitating a crisis, not on a crisis being precipitated solely by removal of a sitting President.

That was Justice White; here is Justice Souter, in the same case:

"If the Senate were to act in a manner seriously threatening the integrity of its results...judicial interference might well be appropriate." Walter Nixon v. United States, 506 U.S. at 253.

Again, both opinions assume a worst case scenario, a case in which the Court has to preserve the integrity of the Constitution because Congress (the Senate) has flagrantly undermined that integrity.  And both depend on the actions of the Senate to prompt a judicial review.  The actions of the House, the very grounds for impeachment, are not covered by either of these arguments.

Which puts us back at the question of what constitutes "high crimes and misdemeanors."  Alan Dershowitz tried to argue the Supreme Court could intervene at what is, essentially, the indictment phase of the process; but that isn't done even in criminal cases.  If the indictment leads to an acquittal, it is proof the indictment was flawed.  If the indictment leads to a conviction, the indictment can still be challenged on appeal, and the grounds for the indictment nullified for all future similar cases.  But I can't think of a way an indictment can be challenged on appeal, any more than a denial of a motion for summary judgment (basically a dismissal) can be appealed before the final judgment is entered.  Interlocutory appeals (appeals before final judgment) are both rare and rarely granted.  The clear impetus of the analysis at FindLaw is that impeachment must result in removal from office (i.e., a final judgment) before the Court could review the impeachment, if it could review the impeachment.  And the Court seems to have understood it could not interfere in that process absent compelling Constitutional grounds, i.e., the failure of the Senate to conduct any trial at all.  It's doubtful the Court would even review the propriety of the trial (especially if the Chief Justice agreed to the conduct of it by overseeing it*), so it would seem the grounds for judicial review, even if possible, are very narrow and specific indeed.

Trump says he'll go to the Court if the House even "tries to impeach."  That's not going to work out the way he thinks it will.

*The CJ's involvement in the trial would probably also prompt a recusal, bringing the matter before a court that could rule 4-4, and so not decide at all.