To start, a simple question: which is the most influential of the three branches of America's government, all "co-equal" according to the constitution? The obvious and correct answer is, of course, the presidency. In domestic policy the occupant of the White House may have less clout than a British prime minister, underpinned by the "elective dictatorship" of a parliamentary majority. But at home and abroad, the president remains the face of his country; by sheer force of personality, he can change history.

But which branch comes next? Not Congress, divided against itself to the point of paralysis, the epicentre of the country's political dysfunction. The answer is the bit you tend to forget about: the judicial branch, or the Supreme Court.

"Equal Justice Under Law," reads the inscription above the colonnade at the entrance to the Neoclassical building, majestic yet oddly modest too, that houses the court. It appears what it is meant to be: the embodiment of impartiality – as John Roberts, the current Chief Justice, puts it in baseball parlance, an umpire who merely "calls balls and strikes", handing down final judgment on what is in accordance with the constitution and what is not.

But matters are no longer that simple. As gridlock has overtaken Congress, which is supposed to make the laws of the land, that role has slipped by default to the Supreme Court, whose members, once nominated by the president and confirmed by the Senate, hold their critically important jobs for as long as they wish, unencumbered by the need for re-election, accountable only to the Almighty and themselves.

Supposedly the court is above politics; in fact, it is now more political than ever, divided into what amount to Democratic and Republican factions, with the latter in a majority.

The turning point was the Bush vs Gore case, when the court, not voters, decided the outcome of the 2000 presidential election. Even then, however, the link between the justices and the political party of the president who appointed them was understood, rather than trumpeted. But now truth has been spoken out loud, by Justice Ruth Bader Ginsburg, in the unlikely setting of an interview with Elle magazine.

Ginsburg has been on the bench some 21 years, a pillar of the court's liberal (that is Democratic) bloc. Some fellow liberals have quietly urged her to resign while the going is good, with a Democrat in the White House who presumably would pick someone of similar persuasions to replace her.

But she refuses, and with remarkable frankness told Elle her reason. "Who do you think President Obama could appoint at this very day? If I resign any time this year, he could not successfully appoint anyone I would like to see in the court." The Senate may have scrapped the filibuster for lower federal court appointments, she noted. "But it remains for this court. So anybody who thinks that if I step down, Obama could appoint someone like me, they're misguided."

So she will soldier on, but her calculation may be wrong. If the Republicans claim a Senate majority and then win back the White House in 2016, Ginsburg's successor would be even less to her liking. And few seriously think she can hang on until 2020 or 2024, when a Democrat might win back the presidency.

And so the current divide on the court will be perpetuated, as the very paralysis of Congress has made the two parties even more concerned with the balance of power within the judicial branch. Like their liberal counterparts, conservative justices will wait until a Republican is in the White House before they step down.

In practice, only an untimely appearance of the Grim Reaper can change things – and maybe not even that. It's unlikely in the foreseeable future that either party will attain the 60 Senate votes need to override a filibuster. It is simply inconceivable that if a conservative justice died with Obama in the White House, Republicans would allow an appointment that would create a liberal majority.

But Justice Ginsburg has inadvertently raised another issue. Should she and her colleagues be entitled to stick around for ever – or at least until they are sure they will be followed "by someone I'd like to see on the court"?

Maybe it's the bracing discipline of constitutional law, or something in the air inside that splendid marble edifice, but Supreme Court members have achieved something close to eternal life. The five most recently retired justices had served an average 25 years each – and given the usual age of appointment (about 50) and the wonders of modern medicine, that stint could grow even longer in future.

Once, moreover, there was a certain unpredictability about the court, as the odd justice expected to be a solid conservative veered left, but no more. The 1991 appointment of Clarence Thomas, by one measure the most conservative justice in 70 years, to replace the civil rights lion Thurgood Marshall shifted its balance decisively to the right, and so it has remained to this day.

Nominated by Bill Clinton and confirmed in 1993, Ginsburg is 81, and has had two brushes with cancer. Three other justices are in their late seventies. No one disputes she's still at the top of her game – as will be agreed by anyone who reads her razor-sharp opinions, or who watches her at oral arguments, peppering lawyers for both sides with questions. But again, do she and her unelected, unaccountable colleagues have the right to engineer the succession of "someone I'd like to see on the court"?