The execution was delayed for more than 3 hours while his attorneys tried to get a stay from the U.S. Supreme Court, CNN reported. They argued police in Texas violated international law because Tamayo was not given the opportunity to get assistance from a Mexican consul when he was arrested.
Tamayo was pronounced dead at 9:32 p.m. in the state prison in Huntsville, corrections officials said.
The Texas Board of Pardons and Paroles rejected a request for clemency earlier Wednesday.
Tamayo, 46, made no final statement before he was executed by lethal injection.
The case attracted attention from the Obama administration and the International Court of Justice. U.S. Secretary of State John Kerry warned Tamayo's execution would be "extremely detrimental" to U.S. interests in a letter he wrote to Texas Gov. Rick Perry in September. Mexican officials have intervened, asking for a stay of execution and for Texas to abide by a 2004 ruling of the International Court of Justice the United States must review the convictions of 51 people sentenced to death.
Tamayo's execution puts at stake "Texas' respect for the Constitution of the United States and for international law," members of the International Commission against the Death Penalty wrote in an opinion piece published Sunday in the Latin Times.
Members of the commission, which is supported by 18 nations, include Bill Richardson, the former governor of New Mexico, and Federico Mayor, former director of UNESCO and Spain's former director of education and science.
Tamayo was convicted of shooting Officer Bill Gaddis, who had arrested him and another man for robbery in 1994.
"It doesn't matter where you're from -- if you commit a despicable crime like this in Texas, you are subject to our state laws, including a fair trial by jury and the ultimate penalty," Lucy Nashed, a spokeswoman for Gov. Rick Perry, said.