Could the Senate simply decide to stop confirming Presidential
nominees to be judges on the Supreme Court, even after there are no
judges left on the Supreme Court?
Probably yes. The Merrick Garland nomination seems to provide a precedent for this kind of action.
I don't believe that it is possible to provide a definitive answer to this question. It would trigger a constitutional crisis (or at least set the scene for a constitutional crisis when a case in the exclusive jurisdiction of the U.S. Supreme Court arises and has no tribunal to resolve it) and in a crisis there is no reliable way to determine what result would follow.
Keep in mind that the absence of a quorum on the U.S. Supreme Court does not constitute the absence of a judicial branch. Federal district courts and federal courts of appeal would continue to function and could rule on any justiciable legal issue presented.
It isn't obvious, however, that a loss of a U.S. Supreme Court quorum due to a failure of the Senate to confirm any appointments or to go into recess allowing a recess appointment to be made on a temporary basis, would give rise to a justiciable issue. It is more likely that the courts would see this as a political question.
But, I wouldn't bet a dime on the federal courts ruling one way or the other if presented with what is largely a question of first impression even if there are a couple of historical analogies.
For example, I could see a court saying that the lack of a discretionary appeal to the U.S. Supreme Court does not violate the constitutional rights of a litigant, but that there is a constitutional right to an alternative remedy of some sort to be crafted by a lower court when the U.S. Supreme Court lacks the appointees to function, in cases within the original jurisdiction of the U.S. Supreme Court or even in cases where there is a non-discretionary right to an appeal under the law. Congress can't deprive someone of due process through inaction. I could also see a lower federal court finding that the lack of a discretionary appeal from a state's highest court violates a due process right to some federal forum, even though no fundamental right is violated by a lack of a discretionary appeal from a federal court of appeals decision.
By analogy, when a state legislature fails to adopt a redistricting plan that complies with the constitution, which is normally a political question, this failure to act allows a federal court to craft a redistricting plan as a remedy for the failure of the legislature to act, in order to protect the constitutional rights of voters and candidates in future elections, even though no express language of the constitution or statute addresses the remedy when a state legislature fails to pass a redistricting plan. I could even imagine a trial court appointing temporary U.S. Supreme Court justices from judges on the existing Article III judiciary, if the U.S. Supreme Court lacked a quorum and someone had a right to U.S. Supreme Court due process, which would serve until the President and Senate filled the vacancy, or until the claimant with the due process right's claim was moot or was resolved.
Alternately, lower federal courts could decide that this presents a conflict between the provisions of the constitution for amending the constitution and those for appointments (as well as the constitutional right of certain litigants to original jurisdiction in the U.S. Supreme Court which often has exclusive jurisdiction of such cases denying due process to someone with such a case and no court to hear it) and find that one or the other is violated either way because the Senate's failure to act was in substance an unconstitutional amendment to the U.S. Constitution by the U.S. Senate acting alone. On that theory, a sensible remedy might be for the court to declare that the U.S. Senate if it fails to act within X days will be deemed to be in recess so that the President may make a recess appointment. What constitutes a recess was recently held to be justiciable in a case involving recess appointments to the NLRB, and this unique situation could be held to be a special and unique exception to the general rule regarding recess appointments set forth in that case. This has the virtue of maintaining separation of powers and of using a close analogy to a situation contemplated in the original constitution by its drafters.
Of course, in light of the current composition of the U.S. Senate and the current President, and the precedent that the "nuclear option" can abolish the filibuster for some kinds of judicial appointments (a parliamentary ruling which is almost surely not justiciable due to express language vesting procedural questions in the U.S. Senate in the Senate and not the courts in the U.S. Constitution), this question is unlikely to present itself any time soon.