Can the GOP legally block Obama's appointment to the Supreme Court?
The short answer to this question is "Yes." This is not exclusive to the Republican party, though. The Senate can block a Supreme Court candidate no matter which party is in the majority. The Constitution provides in Article II that the President appoints with the "advice and consent" of the Senate. This does not require that the Senate approve an appointment, merely that the Senate must consent to the candidate in order for that person to actually become a Supreme Court Justice. Additionally, there is no mandate for the Senate to even hold hearings on a candidate, much less to approve one. Historically, the Senate has declined to approve twelve candidates, most recently in 1987, when it refused to approve President Reagan's candidate, Robert Bork. In fact, this led to a new verb in the English language: to "Bork" someone was to orchestrate an unfair campaign to prevent a justice from being approved by the Senate. Whether that campaign was actually fair or unfair is a judgement that rests in the eye of the beholder.
Whether or not the Senate maintains its present position on this issue, declining to even hold hearings, is going to be a very carefully calculated political position. As it stands now, there are four conservative justices and four liberal justices. This means that many issues are likely to result in a tie vote. When this happens, whatever the court below has decided stands as the law in that circuit. Only when there is a tie-breaking vote does a decision of the Supreme Court become the law of the land and have precedential value. Those of a conservative bent in the Senate will be concerned about more liberal decisions at the circuit level standing as good law, so that is one part of the political calculus. Another aspect Senators must consider is the public's perception of its unwillingness to act, something that has plagued the Senate in recent years and has contributed to the unfavorable ratings it gets from the public, historically, I believe, the lowest of all time. Still another political consideration is the voters back home for each senator. What will their responses be to a failure to consider or appoint a replacement? All of these are what the Senate will consider. I would like to think that they might also consider what their ethical obligations are to the people who elected them as representatives meant to do the work of the United States government. Sadly, that seems to be a rapidly disappearing consideration.
One interesting thing to consider is what would happen if this issue, the legality of the Senate's refusal to act, were to end up in court. The first problem would be the question of standing. Standing is the concept that one must be able to show harm in order to sue in court, so the first question would be who has standing to sue the Senate for this. It might be that the only person who would have standing would be the candidate not acted upon by the Senate. Then we need to ask whether that person would need to sue the Senate as a body or individual senators. Certainly, not all Senators have refused to consider a nominee. It would be difficult to make a case that all American citizens can sue the Senate. These taxpayer suits are usually thrown out for lack of standing, the harm alleged being far too amorphous. Supposing that a nominee were to have standing, this case would go to a federal district court, be appealed to a circuit court, and then end up at the Supreme Court, with its equally divided justices. It is difficult to contemplate how all of this would turn out, particularly because there has been no constitutional violation that could be alleged, and I'm not certain if there is any other legal theory on which a case like this could proceed. Nevertheless, it's an interesting thought experiment!