While this is interesting, I'm not sure that it is answerable. Consider the line item veto. Congress passed a law giving the president a line item veto. Rudy Giuliani (among others) took the law to court and successfully argued that it was unconstitutional. Until the actual court case on this issue, I'm not sure that it is actually known. Of course, it is possible that there is an actual precedent on this issue that is obviously controlling. But I don't happen to know of one.
The actual text of the Constitution for the United States from Article II, Section 2:
[The President] shall nominate, and by and with the Advice and Consent of the Senate, shall appoint Ambassadors, other public Ministers and Consuls, Judges of the supreme Court, and all other Officers of the United States, whose Appointments are not herein otherwise provided for, and which shall be established by Law: but the Congress may by Law vest the Appointment of such inferior Officers, as they think proper, in the President alone, in the Courts of Law, or in the Heads of Departments.
Clearly a law would work for anyone who can be described as an "inferior" officer, but that specifically exempts Supreme Court justices, ambassadors, and "public Ministers and Consuls". It doesn't define ministers and consuls, but I believe that they are normally considered to be the cabinet secretaries and other such heads of departments. It basically says that for other positions, Congress may write law and the law would be binding.
Clearly a constitutional amendment, properly written, could change this, even for officers who are not inferior (in the sense of reporting to a superior that is not the president). They might have to be careful with the wording to avoid the courts narrowing it adversely, as they did with the eleventh amendment. A constitutional amendment is also difficult to pass, requiring the agreement of two thirds of each chamber of Congress and the approval of three quarters of the states.
It is possible that a Senate rule would be sufficient for non-inferior officers. It's not guaranteed, but the courts often defer to the Senate on such things. If the Senate says that a rule is sufficient, then the Supreme Court might allow it. But we can't know for sure until it is tried. The courts could rule that Senate consent must be explicit.
A law would seem to actually have the weakest chance for non-inferior officers. It's not clear that the House has any role in the confirmation of superior officers and there is nothing that gives them one. If the House had that power, we'd expect to see it listed as it is explicitly listed for inferior officers. As such, I'm not convinced that such a law has a constitutional basis. That said, I doubt that there is a binding precedent on that exact question. So the Supreme Court would have considerable leeway in writing up a precedent.
It has been speculated that the Supreme Court could step in and require this. It could. But there is no evidence that it is going to do so. It is far more likely to say that if this is broken, then Congress should fix it. This would not be the first time that the Court found a constitutional requirement where none had previously existed, but neither is that a common occurrence.
What happens if a nominee doesn't show up for Congressional hearings? If the nominee has sufficient support to delay a vote, this reverses things. Now the nominee gets autoconfirmed after the deadline.
Another problem with the Supreme Court making a requirement is that it would only matter in rare situations. This only happens when different parties control the presidency and the Senate but the Senate is close. If the Senate were more solidly occupied, they could simply vote the nominee down. In fact, even with the barest majority, it is possible to vote the nominee down. So a forced vote might not actually improve things.
The best hope would then be that this might be an issue in the next election. But as partisan Democrat Paul Begala said, "No one ever got beat for opposing a Supreme Court nominee." If that's so, then there is no incentive for the Senate to vote for the nominee. Senators can talk about how the Supreme Court forced them to vote. "I might have given the nominee the benefit of the doubt, but the Supreme Court deadline means that I can't get more information. Right now, today, I have to vote no."
Given those problems, why would the Supreme Court intervene? The proposed change is as likely to make the problem worse as better. And there's still the same fix: electing better politicians.
How might it work?
I would expect that Congress would allow itself at least as much time as is typical. Looking at the average confirmation time for the last five presidents' nominees, the least is 44 days, the median (and mode) is 55 days, and the longest average is 87 days. This suggests to me that the requirement would be around two to three months or at least 45 days. I don't see a 30 day requirement as reasonable if appointments normally take 44 or 55 days.
A rule could simply say that all nominees who do not receive a confirmation vote within the time period are presumed confirmed. This is certainly the easiest method. But it is problematic in that it essentially forces a vote. Also, what happens if the nominee stonewalls the Senate? If the nominee can simply avoid a vote for however long, the nominee can take office without facing the Senate. To avoid that, the Senate might have to repeal the rule, which can also be done with a majority vote.
A constitutional amendment could change the system. For example, it might replace recess appointments as well. The president could make a nomination and the Senate would have some period to consider it. If the Senate says nothing, the nominee takes office. But the Senate might be able to come back and hold hearings after that. By a simple majority vote, the Senate could fire the president's nominee.
This handles both the Merrick Garland situation and the recess appointment situation. In both cases, the confirmation is presumed after a period of time if the Senate does not act. And it allows for the possibility of a president trying to abuse the system by preventing the Senate from voting within the deadline. The confirmation is not final until the vote.
More simply, a constitutional amendment could explicitly allow Congress to create a law controlling such matters. Congress could set the deadline and determine what happens if it is missed. The amendment could also repeal the existing text covering nominations. Perhaps it could cover the situation where a President-elect would like hearings to occur before taking office so that nominees could be confirmed on inauguration day.
Congress has wide leeway to allow the appointments of "inferior" officers (those who report to some superior who is not the president). As such, a simple law could definitely allow the president to appoint them without confirmation. And it seems a reasonable extension to allow the Senate to maintain an interest.
There is less leeway with ambassadors, Supreme Court justices, and cabinet secretaries and heads of departments. In my opinion, a law would be questionable. A rule might be allowed. An amendment would work but should be carefully written to avoid ambiguity. A Senate rule is easier to pass than a constitutional amendment. Unfortunately, any such rule could not be litigated until the actual event which would happen after either the presidency or control of the Senate changed.