Right now the Republican leadership are refusing to confirm a Supreme Court nominee in order to keep a Democratic President's nomination from being accepted.
Can the Senate postpone confirming a judge indefinitely?
What if next year we have a republican president, the senate democrats decide to play the same hand, and refuse to confirm a supreme court nominee for the next four years? If you can delay for a year, who's to say you can't delay for four?
Absolutely nothing in law. In practice, the Republicans currently have a stronger hand than Democrats would with a Republican President in 2017. Republicans currently have a majority in the Senate and can thus block a vote from happening. It's unlikely that Senate Democrats will have a majority if there is a Republican President. While the votes are distinct, it would be uncommon for the two to go in opposite directions in the same election.
Republicans can also point out examples of Democrats using similar rhetoric in election years. Democrats wouldn't have the same protection in an appointment immediately following an election. Democrats already face a tough Senate map in 2018 with twenty-three seats up to the Republicans eight (and two more independents who caucus with the Democrats up). In a less extreme case in 2014, Republicans picked up nine seats. And of course if the Republicans have a Senate majority, they could always change the Senate rules to allow a simple majority to confirm an appointment. Harry Reid already gave up the high ground on that by changing the rules for other confirmations.
The only provision that the constitution has for this situation is recess appointments, where the President appoints someone when the Senate is not in session. Even if the Senate went out of session, Obama might avoid doing that. It would weaken his hand at getting Garland confirmed.
Another possible solution would be for Roberts to ask one of the retired Supreme Court justices to sit on the court. There is precedent for that with lower courts. Both Souter and O'Connor have sat on Appeals court panels since retiring. That would provoke a constitutional crisis though, and it's unclear that it would be upheld.
Tactically it also faces the problem that Souter and O'Connor (and Stevens) are more liberal than Roberts, so there is little incentive for him to ask their help in this way. Doing so would tend to lead to him being in the minority in many decisions that are currently tied. There is no current retired conservative justice that he could assign.
In the long term, this problem could be resolved by a constitutional amendment changing how Supreme Court justices are confirmed. It's unclear how exactly the current system would change though. Note that if a vote were held, the Senate could vote against Garland. Or he could be filibustered. Either way, we'd still be effectively in the same spot.
Short answer: Yes.
Longer answer: The president has the power to appoint judges, but only the Senate can confirm them. The senate has in the past sat on nominations for extended periods, although not for the Supreme Court justice. There is no requirement in Article I of the Constitution that the Senate confirm, or even take up a President's nomination. Importantly, there is no requirement that the Supreme Court have 9 people, so technically there isn't an "empty seat" at all, unless the Senate chooses to vote on one. Additionally, as FDRs court stacking showed, there is no reason the President couldn't nominate 2,3, or even 10 people and the Senate just select who they want.
Two other examples of long delays between 'nomination' and confirmation bear mentioning. The oldest signed but unratified treaty by the U.S. according to wikipedia was signed in 1930! The 27th Amendment was passed by Congress at the same time as the Bill of Rights, but because it had no expiration, wasn't ratified by the states until 1992. Another amendment passed at the same time is [still technically pending]! Long durations, therefore, are neither prohibited by law, nor particularly uncommon.
They can go even further than that.
In 1867 Congress was unwilling to accept an Andrew Johnson appointment, so they passed a law reducing the number of justices from ten to seven.
After Grant had taken office they promptly increased it to nine.
The correct answer is NO. At the end of any session of congress(2 years), all presidential appointments expire. The president could renominate and have it sit another 2 years, but there is a expiration date on all nominations. For example see Merrick Garland. His nomination expired officially with that congress.