The U.S. Supreme Court hears the challenge to the law, brought by 26 states, in late March. Supporters of the law want the conservative Thomas to take a hike. Opponents of the law want Kagan home watching college basketball instead of sitting in the courtroom for the healthcare argument.
The move to pry Thomas from the case centers on his wife, who founded and received income from a group dedicated to the repeal of the healthcare reform law.
Virginia Thomas, who worked for the Heritage Foundation and in Congress for then-House Majority Leader Dick Armey, helped found the conservative group Liberty Central in 2009, but has since left it.
Last year, House Democrats sent a letter to the U.S. Judicial Conference, a body headed by Chief Justice John Roberts that oversees the federal courts, complaining about Thomas' failure to list his wife's activities and income on his financial report. Thomas said the errors were inadvertent, but had to scramble to make corrections going back a decade,
The letter, signed by 53 Democrats, called for an investigation.
Earlier in the year, 74 House Democrats urged Thomas not to participate in the in the healthcare challenge case. Showing how scandal curls around to meet scandal in the nation's capital, that group was led by Rep. Anthony Weiner, D-N.Y., who himself had to resign after e-mailing graphic pictures of himself to various young ladies.
The move to oust Thomas is beginning to seem like ancient history as the pace picks up and both sides gear up for argument next month. But the pressure on Kagan is much more recent.
In December, Judicial Watch, which describes itself as "the public interest group that investigates and prosecutes government corruption," hosted a panel discussion on "whether Justice Elena Kagan should recuse herself from the pending Supreme Court case concerning the Patient Protection and Affordable Care Act."
Interestingly, during the Bush administration, the normally right-wing Judicial Watch sued Vice President Dick Cheney, seeking records of oil companies and other energy producers invited by Cheney for closed-door talks on forming the country's national energy policy.
Cheney lost in the lower courts, which ordered release of the records, and he took his case to the U.S. Supreme Court. Months before the high court heard the case, Cheney and Justice Antonin Scalia flew down South on a hunting trip, shooting half-tame ducks on a farm.
Though he flew one-way on Air Force Two, Scalia refused to recuse himself, uncharacteristically giving the reasons why. He and the vice president had never been alone together, he was a longtime friend -- and if justices recused themselves, or withdrew, from cases involving friends in government the Supreme Court would come to a standstill.
With Scalia in the majority, Cheney won his Supreme Court case 7-2.
In the most recent move to oust Kagan, Fox News reports, opponents of the Affordable Care Act "point to a 2010 case regarding a San Francisco health measure, in which then-[U.S.] Solicitor General Kagan's office filed [a friend-of-the-court] brief touting the newly passed healthcare law."
The Fox News report said after Kagan had been nominated to the Supreme Court in May 2010, "Principal Deputy Solicitor General Neal Katyal sent her a memo outlining the cases in which she had 'substantially participated.' Katyal specifically referenced the Golden Gate case, noting it had been 'discussed with Elena several times.'"
That led Heritage Foundation Senior Legal Fellow Hans von Spakovsky to call for Kagan's recusal. "I don't see how any ethical lawyer adhering to professional codes of conduct could not consider that they need to recuse themselves from this case," he told Fox News.
But the Fox News report pointed out other scholars scoff at the idea, saying the issues in the Golden Gate case are distinct from those before the Supreme Court.
"Absolutely different cases, absolutely two different sets of issues, and it does not in any way support the idea that Justice Kagan should recuse herself," Elizabeth Wydra, chief counsel at the Constitutional Accountability Center, told the broadcaster.
The conservative group Freedom Watch filed a request at the Supreme Court last month asking the high court to let it file a friend-of-the-court brief calling for Kagan's recusal, and asking to participate in the upcoming argument.
The Supreme Court denied the request Jan. 23, but added in its order, "Justice Kagan took no part in the consideration or decision of this motion."
Before panic sets in at the White House, Lyle Denniston, the highly respected veteran reporter for SCOTUSBLOG.com, explained the recusal probably came about because the motion "had directly challenged her judicial independence."
He pointed out that she had participated in consideration of all other aspects of the case, and was not signalling she would withdraw from hearing the challenge in court or from voting in a final resolution of the case.
In a Brookings Institution analysis of the dispute, "What's So Hard About Regulating Supreme Court Justices' Ethics? -- A Lot," visiting fellow Russell Wheeler said, "There are no signs that either justice will sit out the case.
"These recusal demands are mostly tactics to try to influence who decides the case or delegitimize the decision, but they reflect a growing debate over whether the justices' ethics need more regulation to avoid conflicts of interest, or their appearance," Wheeler said. "With Gallup reporting the court's approval rating at 46 percent, second lowest since 2000, it's a debate worth having.
"The problem is not that federal judicial ethics provisions only apply to lower federal court judges -- a common but erroneous claim. The problem -- unsolved so far -- is creating mechanisms to regulate the justices' behavior that don't create more problems than they might solve. Some proposals, for example, would suck other federal judges into partisan battles over Supreme Court recusals."
He pointed out the federal statute on judges' disqualification directs any "justice [or] judge . . . [to] disqualify himself in any proceeding in which his impartiality might reasonably be questioned," and lists some situations that would apply. They include owning just one share of stock in a party to the litigation.
But what the federal law doesn't take into account is that the statute is usually enforced by the appellate courts. At the U.S. Supreme Court level, there is no higher court for appeal.
Legislation introduced last March would tell the U.S. Judicial Conference to create such a court to hear appeals from unsuccessful recusal motions and to "decide whether the justice ... should be so disqualified."
But such a court, Wheeler said, likely would violate the Constitution's mandate of "one Supreme Court."
The federal Ethics in Government Act requires a "willful" failure to report gifts or income -- remember Thomas' explanation for his missteps? The U.S. Judicial Conference's Code of Conduct applies to judges but doesn't really apply to high court justices.
"The absence of formal review mechanisms for justices' ethical decisions is a necessary imperfection in the system," Wheeler contends. "The frustration behind recent proposals to establish such mechanisms is understandable, but those proposals would likely create more problems than they would solve."
Wheeler suggests several measures to help the high court's credibility.
-- More transparency, citing Scalia's extensive 2004 memorandum on why he wasn't withdrawing from the Cheney case. Almost always, justices withdraw from a case without explanation.
-- The justices themselves could adopt and make public a formal set of standards to guide them on when recusal is warranted.
-- If the justices, behind closed doors, already have adopted resolutions regarding the various ethics acts or the conference's Code of Conduct, they should be made public.
-- The justices should "master the admittedly arcane web of statutes and policies that govern and guide them," so they can answer questions before Congress.
Notable deaths of 2014 [PHOTOS]