A day after prosecutors dropped a felony charge against Gov. Eric Greitens, Missouri lawmakers had a message for the governor on Tuesday: Don’t celebrate quite yet.

Mr. Greitens has, for now, survived a legal battle over a felony invasion of privacy charge stemming from accusations that he took an explicit photograph of a woman with whom he had an affair, without her consent. But in the Republican-dominated Missouri Legislature, where Mr. Greitens, a Republican, has few friends and many adversaries, the threat of impeachment has only intensified.

Legislators declared that the dismissal of the criminal charge against Mr. Greitens would not deter them in their plan to continue investigating the governor and, if necessary, remove him from office, something no Missouri Legislature has done before. On Friday, the Missouri House will gavel in for a monthlong special session to consider disciplining the governor. Sometime in the next several weeks, it is likely that the chamber will vote on impeachment.

That timeline was unchanged after the decision on Monday by Kimberly Gardner, the St. Louis circuit attorney, to drop the invasion of privacy charge. A bipartisan House committee that has been investigating Mr. Greitens for months could decide at any time to make a recommendation on whether to impeach the governor. If a majority of House members voted to impeach, the State Senate would select a special commission of judges to try the case.