Cuccinelli said that conflicting court decisions about the law's constitutionality have created sufficient uncertainty about implementation of the sweeping law to justify speeding Supreme Court review.
The Justice Department will oppose the motion, saying that the case should be fully heard by lower courts before the Supreme Court takes action.
The high court has granted such requests infrequently, and many experts said they think Cuccinelli's filing is a longshot. Supporters of the law said that the provision at the heart of the legal dispute - a requirement that individuals buy health insurance - will not go into effect until 2014.
A U.S. District Court judge in Virginia ruled in December that it is unconstitutional to require people to buy health insurance, as envisioned in the law.
The federal government appealed, and the Fourth Circuit Court of Appeals is scheduled to hear the case in May.
A federal judge in Florida ruled this week in a suit filed jointly by Florida and 25 other states that the law is unconstitutional. In other cases, two other federal judges have said the law is constitutional.
According to court rules, petitions to bypass appellate review are granted only in cases that are of "such imperative public importance" that they require changing normal procedures.
Cuccinelli will argue that conflicting opinions over a law that will reshape one of the largest sectors of the economy justify the speedy review.
"Regardless of whether you believe the law is constitutional or not, we should all agree that a prompt resolution of this issue is in everyone's best interest," he said in a statement.
But a spokeswoman for the Justice Department said that the expedited review would not significantly change the case's timeline, since it is to be heard in May, likely allowing the Supreme Court to take up the case during its next term.
"The Department continues to believe this case should follow the ordinary course of allowing the court of appeals to hear it first so the issues and arguments concerning the Affordable Care Act can be fully developed before the Supreme Court decides whether to consider it," spokeswoman Tracy Schmaler said in a statement. "Virginia's suit is based on a state statute that is not applicable nationwide."