State Rep. Austin Scott unveiled the resolution on Crossover Day, the deadline by which legislation is supposed to pass at least one chamber to become law this session. Scott, a Republican candidate for governor, said the bill has 33 GOP sponsors, including House Speaker David Ralston.
Scott said he believes the health care legislation is "fundamentally unconstitutional."
"I wish to make the will of the people of Georgia abundantly clear to our own attorney general," Scott, of Tifton, said.
Gov. Sonny Perdue had asked Thurbert Baker to sue over the health care bill. On Wednesday Baker refused, saying such a lawsuit stood little chance of succeeding and would be waste of money for the cash-strapped state.
Baker is a Democratic candidate for governor.
Perdue fired back Thursday saying he would appoint an outside lawyer - who had volunteered to work for free - as a special attorney general to file the suit.
Scott said Friday that believes the attorney general has "a constitutional responsibility" to pursue the legal challenge himself. He acknowledged the measure is racing against the clock on Friday.
Earlier Friday, the House failed to muster enough votes to suspend its rules so that a related bill could be considered. That bill would have limited the power of the attorney general. The rules suspension fell 10 votes short of the 120 supermajority needed.