Hatch's call is part of the broad legal and political maneuvering on both sides for the most favorable conditions surrounding court review of President Barack Obama's signature domestic policy accomplishment.
His comments came the same week that Virginia Attorney General Ken Cuccinelli said he plans to file a motion to take the case directly to the Supreme Court, bypassing an appeals court, after he won a federal judge's ruling in December against the law's requirement that most Americans buy health insurance.
On Monday, a second federal judge declared the law unconstitutional. Two other judges have upheld it.
Hatch said he is sure that Kagan participated in discussions about the law and challenges to it while she served in the Justice Department as Obama's top Supreme Court lawyer. Hatch told Fox News that he believes Kagan "should recuse herself," although he noted the justice alone will make that determination.
The Utah senator also voted against Kagan's confirmation to the Supreme Court in August.
The issue of Kagan's participation looms large if the justices' views on the health care law divide along ideological lines. Her absence in such a situation could leave the court split 4-4, which would prevent it from settling the subject with a uniform set of rules for the entire country.
Kagan addressed her participation during her confirmation hearing. She said then that she "attended at least one meeting where the existence of the litigation was briefly mentioned, but none where any substantive discussion of the litigation occurred." Kagan left the administration in August, about five months after the health care overhaul became law.
A federal law sets conditions for when judges who formerly worked for the government should bow out of cases.
Stephen Gillers, an ethics expert and law professor at New York University, said he thinks Kagan can take part in the case if her involvement was limited and did not include offering advice about how to defend the law.
"She should not be sitting in the matter if she is essentially reviewing her own advice as a government lawyer," Gillers said.
Douglas Kendall, president of the liberal interest group Constitutional Accountability Center, said he believes Kagan already has made clear she is fit to hear the health care case. "I think Sen. Hatch is wildly off-base suggesting that she will or should recuse in this case," Kendall said.
The law governing judges' participation was strengthened after William Rehnquist refused to stand aside from a Supreme Court case involving allegations of the Army spying on U.S. citizens, even though he had expressed an opinion about the surveillance as well as the actual lawsuit while serving in the Justice Department.
The other issue of immediate concern regarding the health care law is how quickly the case might reach the Supreme Court. Cuccinelli said the case should go directly to the high court to spare states and companies expenses for a law that might be ruled unconstitutional.
That also would prevent the federal appeals court in Richmond, now dominated by judges appointed by Democratic presidents, from considering the case.
The Obama administration said the challenge should follow the ordinary path through an appeals court. Two other appeals courts, in Atlanta and Cincinnati, where Republican appointees hold the majority of judgeships, also will be considering challenges to the law.
Justice Ruth Bader Ginsburg was asked Thursday night about the law's route to the Supreme Court. She did not respond directly, but said only rarely do the justices agree to step into a lawsuit before appeals court judges have weighed in.
She said that one benefit of the slower process is that the issues get more fully developed for presentation to the Supreme Court.
"We do so much better when we have the views of those federal judges," Ginsburg said
The relatively few occasions in the past 60 years when the court has agreed to hear cases before appellate judges have weighed in include President Harry Truman's seizure of steel mills during the Korean War and President Richard Nixon's refusal to turn over the Watergate tapes to a special prosecutor.