"I personally believe she should recuse herself. I'm sure she participated in discussions at the White House. Participated in discussions in the solicitor general's office. These issues were brought up throughout the process," Hatch said on Fox News last week. "That's going to be up to her what she does."
Kagan has recused herself from several cases since she was appointed to the Supreme Court last August.
Rhode says in a case of this magnitude, all justices -- whether conservative or liberal -- will need to look closely at their records, as the credibility of the Supreme Court could be affected for years to come.
"In a case of this enormous national importance where you're already going to have public concerns about ideological concerns," she said, "you want to be particularly careful that there's no possible personal interest that could be attributed to the justice that might interfere with an objective appraisal."
It remains unclear when the Supreme Court will take up the health care case, although experts say it could come within the next year.
On Tuesday, Virginia Attorney General Ken Cuccinelli -- who won the case on the district level -- filed papers to bypass the appellate courts and asked the Supreme Court consider the constitutionality of the law.