Adding my own answer because I think I disagree with the current top-vote getter on somewhat pedantic grounds. Bobson's answer is a defensible response to what SCOTUS would have said about their authority to overrule McConnell. But your question was whether SCOTUS had the authority to rule on it at all. To which the answer is basically, sure, if the Obama administration had pressed the issue.
The Supreme Court's authority begins and ends in large part wherever the Supreme Court decides it does. But there are two major checks to the breadth of their authority. First, there has to be a real "case or controversy" to discuss. So the Court can't just rule tomorrow that X, Y, or Z is the law, someone being harmed has to bring a case and ask them to do something. Secondly, Congress has the power to strip the Court of appellate jurisdiction of certain controversies. So if SCOTUS starts to overstep, they can pass a bill saying the Supreme Court has no authority to hear certain cases. (This is only done very rarely.)
In this case, there was definitely a real controversy and Congress didn't pull the rug out from under the Court by jurisdiction stripping. So yes, I think someone could've sued and eventually argued it all the way up to SCOTUS.
That said, would they have the authority is a different question from would they choose to exercise authority. The Supreme Court often rules that certain controversies are non-justiable "political questions" that are poorly suited for the courts to decide, and that the other branches need to work out themselves.
That said, they did weigh in NLRB v. Noel Canning, which at its heart was a question of interpretation of the Senate's power to decide when its in recess and whether the president could declare unilaterally that a recess was happening. That's pretty superficially similar to the heart of the dispute of the Garland nomination, which revolved around dueling interpretation of whether or not the Senate had given "advise and consent."
Of course, Noel Canning is probably a window into why the Obama admin didn't sue... they got spanked 9-0. But SCOTUS in that case rejected the interpretation that the underlying controversy itself was a political question.
I guess the tl;dr here is SCOTUS had the authority to rule on Garland appointment, but the smart money is they would've either a. disclaimed the authority to do anything about or b. ruled outright in favor of McConnell's interpretation