“With today’s disappointing and misguided political decision, my confidence in our prosecutorial system is shaken, but not broken,” Mr. Greitens, 43, wrote on Facebook. He suggested that the charges were the work of a “reckless liberal prosecutor who uses her office to score political points,” an apparent reference to Kimberly M. Gardner, the St. Louis prosecutor whose office presented the case to the grand jury and who is a Democrat.
A lawyer for Mr. Greitens, Edward L. Dowd Jr., said in a statement that the charge was baseless and that he would be filing a motion to dismiss the case.
Mr. Greitens showed no immediate signs of quitting, but he was certain to face pressure to leave, and some lawmakers said they were pushing for an investigation, which could be a first step to seek impeachment.
Though his fellow Republicans hold large majorities in the State Legislature, the governor has frequently clashed with members of his own party. State Senator Robert Schaaf, a Republican, said lawmakers should “move swiftly to bring this to a resolution” and possibly impeachment if the governor did not resign.
Republican leaders in the State House raised their own doubts about Mr. Greitens’s future.
“We will carefully examine the facts contained in the indictment and answer the question as to whether or not the governor can lead our state while a felony case moves forward,” a joint statement from the Republican leaders, including Todd Richardson, the speaker of the House; Elijah Haahr, speaker pro tem; and Rob Vescovo, the majority floor leader, said. “The people of Missouri deserve no less. We will begin the process of tasking a group of legislators to investigate these serious charges.”
Ms. Gardner said the statute of limitations for charges to be filed would have expired next month.
“As I have stated before, it is essential for residents of the City of St. Louis and our state to have confidence in their leaders,” Ms. Gardner said in a statement. “They must know that the Office of the Circuit Attorney will hold public officials accountable in the same manner as any other resident of our city. Both parties and the people of St. Louis deserve a thorough investigation of these allegations.”
Mr. Dowd said the governor had offered to meet with Ms. Gardner, but that she refused and “proceeded to file an indictment that has no facts.” Mr. Dowd said his client would cooperate if a legislative committee investigates.
Last month, after the claims against Mr. Greitens became public, Mr. Greitens acknowledged an affair and asked for forgiveness.
“This was a deeply personal mistake,” Mr. Greitens and his wife, Sheena, said in a joint statement. “Eric took responsibility, and we dealt with this together honestly and privately.”
The tawdry political drama is taking place in a state where Senator Claire McCaskill, a Democrat, is locked in what is expected to be one of the most hard-fought Senate races in the country.
Mr. Greitens was widely believed to have aspirations for higher office, perhaps even the presidency, and had appeared in Iowa, Michigan and Virginia in recent months in an apparent effort to burnish his national brand.
He had been well known within military circles. In 2007, after coming home from Iraq, Mr. Greitens started what became known as the veterans empowerment movement when he founded The Mission Continues, a nonprofit that sought to encourage veterans to take on new missions of service in their communities. In 2013, Time magazine named him one of the 100 most influential people in the world.
By Thursday night, lawmakers were raising pointed questions about Mr. Greitens’s future.
Ryan Silvey, a former Republican state senator whom Mr. Greitens appointed to the state’s Public Service Commission last month, said it would be difficult for the governor to survive.
“He doesn’t really have a lot of deep relationships in the Legislature to begin with,” Mr. Silvey said. “I don’t see how he can effectively govern in the current situation. I think that it would probably be best for the party and for the state if he were to resign.”
It was hard to see the prosecution as political, he said, given the fact that the indictment came from a grand jury.
“With it being a grand jury, that tells me evidence was presented to other people who saw that an indictment would be reasonable,” he said.
Under Missouri law, a member of the State House can file a resolution seeking an investigation of the governor, which would then be referred to a committee. That committee could then advance articles of impeachment to the full House for a vote. And if a majority of House members supported impeachment, the State Senate would select seven judges to conduct a trial.
No Missouri governor has ever been removed from office through impeachment.
T.J. Berry, a state representative from Kansas City, said he hoped the Assembly would open its own investigation “to remove the question of political taint” in Ms. Gardner’s prosecution.
While Mr. Berry, a Republican, said that the allegations still needed to be proven, the charges were not a good sign for the governor.
“If you get a grand jury to indict you, there’s probably some proof,” Mr. Berry said.
State Senator Jamilah Nasheed, a Democrat, said “there’s a very dark cloud over the state of Missouri and I truly believe that the governor cannot lead” while under felony indictment.
If Mr. Greitens does not resign, “the people here in the state of Missouri need to rise up and call for his impeachment,” Ms. Nasheed said. “This is a big embarrassment to the state.”
Mr. Greitens, who was permitted to travel throughout the United States as a provision of his release, is scheduled to appear in court on March 16.
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