Bourgeois was pronounced dead at 8:21 p.m. Eastern Standard Time by the Vigo County Coroner.
He was convicted in 2004 of murdering his two-and-a-half year old daughter Jakaren on a military base in Texas.
"Jakaren lost her life to a monster who lived for 18 years after the crime. A child should not have to endure what she did then, now we can start the process of healing," the toddler's family said in a statement.
"I ask God to forgive all those who plotted and schemed against me, and planted false evidence, I did not committ this crime," Bourgeois said in his final statement moments before receiving a fatal dose of Pentobarbital.
Bourgeois' attorney say his client has an IQ of 75, and exeucting him would be unconstitutional.
"It's shameful that the federal government executed Mr. Bourgeois without fair consideration of his intellectual disability," said Bourgeois attorney Victor Abreu.
Bourgeois is the 10th inmate executed since the Trump Administration resumed federal executions in July. Brandon Bernard was executed in the same facility Thursday night.
The Associated Press contributed to this report.
