Native American leaders said the victims are members of the Northern Arapaho tribe and called for a federal hate crime investigation.
Roy Clyde, an employee of the Riverton parks and recreation department, walked into the Center of Hope drug and alcohol detox center on Saturday and shot the men at close range, Riverton Police Capt. Eric Murphy said.
Clyde, 32, left his handgun and shirt on a desk inside and waited outside with his hands up until police arrived, Murphy said.
"He deals with the homeless population every day, and basically he just said that he was tired of watching them urinate and defecate and having sex and vandalizing the city park," Murphy told CNN affiliate KCWY
. He said Clyde reported the shooting to police.
Clyde told police "his decision was not race-based and that he was targeting transient people regardless of race. He specifically indicated that if he had encountered white people meeting his criteria, he would have killed them as well," according to a police affidavit quoted by County 10, a website covering events in Fremont County, Wyoming.
According to the affidavit, Clyde said he first went to Riverton City Park but didn't find anybody to kill "who met his specific criteria," so he went to the Center of Hope because "he knew he would find people meeting his criteria," County 10 reported.
"Clyde stated during the interview that his intent was to kill and that he intended to kill as many people meeting his criteria as he could," County 10 reported, quoting the affidavit.
Many of the people Clyde allegedly complained about -- whom he referred to as "park rangers" -- are Native Americans, Murphy said. It was unclear whether the victims were actually homeless.
Riverton is a town
of about 10,000 people surrounded by the Wind River Reservation
, 2 million acres in central Wyoming that is home to over 3,900 Eastern Shoshone and 8,600 Northern Arapaho enrolled tribal members.
Clyde was charged with one count of murder and one count of attempted murder. Any hate crime prosecution would have to be on the federal level because Wyoming has no hate crime law, Murphy said.
"In my mind at least, clearly there is some hate involved," Police Chief Mike Broadhead told KCWY.
Clyde remains in jail without bond. CNN could not determine if he has a lawyer.
Richard Brannan, a member of the Northern Arapaho Business Council, said the civil rights division of the U.S. Justice Department needs to investigate, according to a tribal statement.
"The council has received a steady stream of reports about abusive behavior towards native people," he said. "In 2013, a Northern Arapaho woman was shot in the face by a passing car in Riverton, and the Riverton Hospital released her without treatment. ... I've lived here my whole life, and the anti-Indian sentiment seems to be getting worse."
Council Member Ronald Oldman said the killing of nine people in Charleston, South Carolina
, "can serve as an example as our community begins to process what has happened here. Like in Charleston, people need to acknowledge that racism played a role in this."
Police didn't identify the victims, but the statement by the Northern Arapaho tribe identified them as tribal members Stallone Trosper and Sonny Goggles.
A fundraising effort is underway
to help the family of Goggles, who remains hospitalized.
When the shooting started, employees at the Center of Hope quickly moved clients into a bathroom for safety, KCWY reported.
Heath Steel of the Volunteers of America Northern Rockies, which operates the center, said the organization will strive to stay positive.
"This was a crime against our family, we're going to come out stronger, we're going to heal, and we're going to ensure that every bit of work we do is to move forward form this in a positive light," Steel told KCWY.