Police were dispatched to a home in south San Antonio on Sunday night and found the wounded 59-year-old near the street.
"The officers attempted life saving measures until EMS arrived. EMS pronounced the victim deceased," San Antonio police said in a statement.
Joss' husband, Tristan Kern de Gonzales, confirmed the actor's death to The Associated Press in a text.
The two were married earlier this year on Valentine's Day.
In a post on X, San Antonio police did not immediately provide any information on what prompted the shooting.
San Antonio police said its investigation "has found no evidence whatsoever to indicate that Mr Joss's murder was related to his sexual orientation".
"We take such allegations very seriously and have thoroughly reviewed all available information. Should any new evidence come to light, we will charge the suspect accordingly," police said.
In a statement, de Gonzales said he and Joss had previously faced harassment, much of it "openly homophobic".
Joss' husband said the person who killed the actor yelled "violent homophobic slurs" before opening fire.
"He was murdered by someone who could not stand the sight of two men loving each other," de Gonzales said.
Before the shooting, de Gonzales and Joss were checking mail at Joss' home, which had been heavily damaged during a January fire that claimed the lives of their three dogs. A man approached the two and threatened them with a gun, de Gonzales said.
"Jonathan and I had no weapons. We were not threatening anyone. We were grieving. We were standing side by side. When the man fired Jonathan pushed me out of the way. He saved my life," de Gonzales said in a statement.
After the shooting, authorities arrested 56-year-old Sigfredo Alvarez Ceja and charged him with murder in Joss' death.
Joss, who grew up in San Antonio, was best known as the voice of John Redcorn, a Native American character on the popular King of the Hill animated series that ran for 13 seasons from 1997 to 2008.
A reboot of the show is set to start in August.
Joss also had a recurring role on the television show Parks and Recreation, playing Chief Ken Hotate. He appeared in two episodes of the series Tulsa King in 2022.
Before he was fatally shot, Joss had been in Austin, located about 130km northeast of San Antonio, for events related to a sneak peak of the King of the Hill revival.
On Saturday, Joss had posted a video on Instagram in which he said he was signing autographs at a comic book store in Austin.
"The fans get to revisit King of the Hill again, which I think is an amazing thing because it's a great show," Joss said in the video, adding he had already done voice work on four episodes of the revival.
Joss' husband said Joss was grateful for his fans.
"To everyone who supported him, his fans, his friends, know that he valued you deeply. He saw you as family," de Gonzales said.
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