Democrats counter that the lawsuit is a political stunt designed to improve Republican chances in the November elections.
A vote on the measure is expected early next week in the House, shortly before a scheduled month-long vacation begins for lawmakers.
Republicans rejected numerous Democratic attempts to change the legislation.
At the same time they decided to require a public accounting of any government money spent to hire outside lawyers or experts in connection with the case.
Republicans have long claimed that Obama has selectively enforced the health care law, pointing to a series of executive orders he has issued since its enactment. The administration disputes the allegations.
The legislation cleared by the committee allows the House to sue any executive branch officials, including the president, for failing to carry out their duties under the Constitution in connection with implementation of the law.
Disputing Democratic assertions, House Speaker John Boehner has vociferously disputed charges that the legal move is politically motivated, or that it is designed to tamp down sentiment among tea party supporters for impeachment proceedings against Obama.
For their part, Democrats have used the possibility of a lawsuit to raise campaign contributions in advance of the November elections with control of Congress at stake.