No, he could not. Not without a case before him. He could have refused to adminsiter the oath of office, but they could have just substituted another justice, or anyone at all really. Maybe the Imam of the DC mosque attended by the 9-11 hijackers and "Major" Hassan. The Constitution does not indicate who, if anyone, should administer the oath. In the early days, the oath was not always administered by the Chief Justice, that is a more recent tradition with no foundation in the Constitution.
Meanwhile to have so refused would likely have created a conflict of interest should a case eventually come before the Supreme Court and Chief Justice Roberts.
This question is, at its root, a nonjusticeable political question. ALL of the political branches, including the Electoral College, the Congress including ALL Republicans and the Republican Vice President/President of the Senate had the opportunity to object to Obama's certification and they did not do so.
So your position is that if somehow the electorate picked the Yemeni born son of Osama Bin Laden and no one objected in Congress, it would be a "done deal", and no action by the Courts would be possible. Yes that is extreme, but it is no different in principal than any other reason for ineligibility being overlooked by the "Political Branches".
Yes, that's my position.
Just a quick followup: Is it your position that elections are subject to post-hoc review by Article III courts?
Is that power one that you are comfortable with courts exercising?