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The Constitution doesn’t give a power to any specific branch for establishing the eligibility of the president. Again, there’s already precedence where the court has reviewed eligibility of a Congressional electee. There’s no constitutional restriction from the court performing the same review of presidential electee or fraudulently seated president.

The current High Court has denied hearings to any of the eight Obama eligibility appeals that have reached them for “cert conferences.” The question remains at the lower court levels as to who has legal standing to bring a lawsuit on eligibility. Thus far in 73 attempts, no one has overcome the legal standing hurdle.

My guess on the Supreme Court of the United States level is that Justice Roberts has recused himself from all Obama eligibility appeals since he swore Obama in (twice) and invited Obama and Biden over to the Court for cocktails and conversation after the Inauguration.

Justice Scalia is a pretty strict “jus soli” judge. Having argued that position in open court in oral arguments in an immigraation case.
In the oral arguments for “Tuan Anh Nguyen v. INS” (No. 99-2071), Justice Scalia made it clear that his view is that natural born citizenship, the requirement to be president, is based on “jus soli” (birth in the United States).

Here is the relevant section from the transcript:
Justice Scalia: … “I mean, isn’t it clear that the natural born requirement in the Constitution was intended explicitly to exclude some Englishmen who had come here and spent some time here and then went back and raised their families in England?

They did not want that.

They wanted natural born Americans.”

[Ms.]. Davis: “Yes, by the same token…”

Justice Scalia: “That is jus soli, isn’t it?”

[Ms.] Davis: “By the same token, one could say that the provision would apply now to ensure that Congress can’t apply suspect classifications to keep certain individuals from aspiring to those offices.”

Justice Scalia: “Well, maybe.

I’m just referring to the meaning of natural born within the Constitution.

I don’t think you’re disagreeing.

It requires jus soli, doesn’t it?”

That leaves Justices Alito, Thomas and Kennedy to possibly be willing to entertain a natural born citizen appeal and it takes four justices (the Rule of Four) to agree to hear a case before the full Court.
I doubt if Sotomayor, Kagan, Breuer, or Bader Ginsberg is interested.