The legislation is headed to Obama to sign, which could happen Tuesday.
"We plan to file the moment Obama signs the bill," Abbott said in a posting on Facebook. Attorneys general for several other states indicated Monday they would join the lawsuit.
House members voted 219-212 late Sunday to approve the health care overhaul that would extend coverage to 32 million uninsured Americans. It also would significantly expand Medicaid, the federal-state health care program for the poor; place new federal regulations on the insurance industry; and allow parents to keep children up to age 26 on their family insurance plans.
Most Americans would have to buy insurance or face penalties. There would be subsidies to help families with incomes of up to $88,000 a year pay their premiums.
Prominent Texas Republicans, including Abbott and Gov. Rick Perry, have been outspoken against the Democratic-backed health care measure. Perry called it a "gross federal overreach."
"Unfortunately, the health care vote had more to do with expanding socialism on American soil than it does fixing our health care finance and delivery systems. The Obama health care bill undermines patient choice, personal responsibility, medical innovation and fiscal responsibility in America," Perry said in a prepared statement.
No Republican joined House Democrats in voting for the bill, and one admitted Monday that he shouted "baby killer" during a speech Sunday evening by a Democratic colleague whose vote was crucial to the overhaul's passage.
Rep. Randy Neugebauer, who represents about two dozen West Texas counties, said he yelled "it's a baby killer" in reference to an agreement reached between President Barack Obama and anti-abortion Democrats led by Rep. Bart Stupak. The third-term congressman said he apologized to Stupak and regretted that his outburst was misinterpreted as a direct reference to the lawmaker.
One Texas Democratic congressman, Rep. Chet Edwards of Waco, voted against the legislation.
Congressional Democrats from Texas who backed the measure said in a joint statement: "This pivotal moment in our nation's history will deliver affordable, accessible health care coverage to all Americans, and this plan will take a stand against rising premiums in Texas which have doubled over the past decade."
The Texas AFL-CIO hailed the bill's passage. The group's president, Becky Moeller, criticized Texas Republican leaders for "just sloganeering over 'states' rights."
"And a leadership that would sue to stop health care reform at the drop of a hat might want to explain its priorities," she said in reference to Abbott's threatened lawsuit.
The nonpartisan Congressional Budget Office said the legislation would cut deficits by an estimated $138 billion over a decade. The legislation includes more than $400 billion in higher taxes over a decade and cuts more than $500 billion from planned payments to hospitals, nursing homes, hospices and other providers that treat Medicare patients.