Others have taken a more nuanced view. They said the government’s response to Ebola in Dallas has been problematic at the local, state and federal level, with confusion and missteps among all agencies and officials. And they said Mr. Perry’s decision to leave for Europe came before the second nurse, Amber Joy Vinson, had gone to the emergency room feeling ill and the news about her air travel between Dallas and Cleveland spread alarm throughout Dallas and the country.

“In retrospect, this has been a pretty rapidly evolving situation that’s gotten much more complicated than most people envisioned at the starting point,” said James Henson, director of the Texas Politics Project at the University of Texas at Austin. “This is the kind of situation that the cliché ‘hindsight is 20/20’ was made for. If he knew then what he knows now he almost certainly wouldn’t have gone to Europe.”

Ebola has arrived in Texas at a complicated time for Mr. Perry politically. His tenure as the longest-serving governor in the history of the state ends in January, 14 years after he first took office. His every move or remark is analyzed as he presents himself as a potential presidential contender. And he has been fighting a legal battle in court, after a Travis County grand jury indicted him in August on charges of abusing his official capacity and coercing a public servant, in a messy case involving his veto power as governor and Austin’s top prosecutor.

After the Oct. 1 press conference, Mr. Perry played a largely hands-off role in Dallas itself in the immediate days that followed, apparently keeping in communication with state agency leaders but having limited direct contact with some of the local officials. Mr. Browne, the governor’s spokesman, said the list of officials the governor had spoken with included Mr. Obama; Dr. Thomas R. Frieden, director of the federal Centers for Disease Control and Prevention; Sylvia Mathews Burwell, the secretary of health and human services; and Dr. David L. Lakey, the commissioner of the Texas Department of State Health Services. No city or county leaders were included in the list Mr. Browne provided.

Although the first three cases of Ebola diagnosed in the United States have occurred in Texas in the span of two weeks, the main public faces of the government’s Ebola response have been local and federal but not state — Mike Rawlings, the mayor of Dallas; Clay Jenkins, Dallas County’s chief executive; and Dr. Frieden with the C.D.C. The state’s primary public figure in the Ebola drama has so far been not so much a face but a voice: Dr. Lakey, the state health commissioner, who appears with Dr. Frieden on the C.D.C.’s frequent news conferences via phone from Texas.

Supporters say that before he left for Europe and after he returned, Mr. Perry has been active in coordinating state resources and personnel. He created a 15-member task force on infectious disease preparedness, made up of experts who are developing a comprehensive state plan and are holding their first hearing in Austin on Thursday. The task force has made initial recommendations, including establishing two Ebola treatment centers in Texas.