It's a great question, one deserving of more coverage and discussion.
Let's break down the two questions into their component pars.
1a. Can Congress be sued? Likely, yes. Of course specific members may be sued and/or tried under impeachment.
1b. Can Congress be sued for blocking any nomination? Possibly yes.
The president has sole power to nominate, and with advice and consent of the Senate, to appoint Supreme Court justices. The actual wording: "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."
To put a little flesh on this - and to get closer to the question of whether the Senate HAS to act - we can look at what our Founders had to say about it. They thought this a relatively straightforward matter. Here are the words of James Iredell, on of the first Supreme Court justices (appointed by Washington), during a debate in his native North Carolina on adoption of the Constitution: ""As to offices, the Senate has no other influence but a restraint on improper appointments. The President proposes such a man for such an office. The Senate has to consider upon it. If they think him improper, the President must nominate another, whose appointment ultimately again depends upon the Senate."
Washington himself wrote (in accordance with Jefferson and John Jay) that Senate powers "extend no farther than to an approbation or disapprobation of the person nominated by the President, all the rest being Executive and vested in the President by the Constitution."
And Hamilton argued nominees should be rejected only for "strong and special reasons."
Also relevant to this question is the Oath of Office taken by senators:
"I do solemnly swear (or affirm) that I will support and defend the Constitution of the United States against all enemies, foreign and domestic; that I will bear true faith and allegiance to the same; that I take this obligation freely, without any mental reservation or purpose of evasion; and that I will well and faithfully discharge the duties of the office on which I am about to enter: So help me God."
The Constitution states: "Senators...shall be bound by Oath or Affirmation, to support this Constitution..."
The case could therefore be made that senators have deliberately evaded a sworn, constitutional duty to consider - in good faith - Supreme Court nominees. The case at least can be made.
1c. Could such a case reach the Supreme Court? Unlikely, no. In theory the Supreme Court could hear such a case, but in general the court views these as political cases - to be decided by voters who may or may not decide to vote the senators out of office. But then again we have been surprised before in recent years when the Supreme Court had the option of referring a political case to the House of Representatives but instead did not (Bush v. Gore, 2000).
2. Are there other angles to get around this deadlock? No. No constitutional ones. Nothing prevents Democrats from pulling the same dirty trick when the shoe is on the other foot, and who is to say it couldn't go on for four years. At that point - or in the event of continued blocking by a Republican Senate - we may either see the Supreme Court simply become smaller and smaller over time as sitting justices step down or die, or, at some point, this slow burn constitutional crisis will boil over. A nice resolution would be something along the lines of an amendment (hah! fat chance!) stating the Senate shall advise President of nominees in timely fashion, and so on, with time limits for high level appointments.