"This is the most offensive thing I've ever heard," Abbott, a Republican, wrote on Twitter. "No one should ever question the loyalty or professionalism of the Texas National Guard."
The harsh riot at the Capitol earlier this month led officials in the city to take severe actions to ensure the inauguration is safe.
Roughly 25,000 National Guard troops have been sent to the capital -- including about 1,000 from Texas.
"I authorized more than 1,000 to go to DC," Abbott wrote. "I'll never do it again if they are disrespected like this."
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U.S. defense officials said they were concerned about an insider attack or other threat from service members involved in obtaining the inauguration, urging the FBI to vet all National Guard troops arriving to Washington D.C. for the event.
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Army Secretary Ryan McCarthy told the AP that officials were informed of the potential hazard, and he advised commanders to be on the lookout for any obstacles within their ranks as the inauguration approaches.
"We’re continually going through the process, and taking second, third looks at every one of the individuals assigned to this operation," McCarthy said.
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He and other leaders say they haven't seen proof of any threats and officials noted that the vetting hadn't signaled any issues that they were aware of.
Acting Secretary of Defense Chris Miller, in a statement on Monday, said that the Department of Defense has also not got any intelligence suggesting a potential insider threat.
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Still, Miller added that the department is "leaving no stone unturned in securing the capital."
"This type of vetting often takes place by law enforcement for significant security events. However, in this case the scope of military participation is unique," he said.
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McCarthy said in an interview after he and other military leaders went through an exhaustive, three-hour security drill in preparation for Wednesday’s inauguration. He said Guard members are also getting training on how to identify potential insider threats.
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About 25,000 members of the National Guard are streaming into Washington from across the country — at least two and a half times the amount for previous inaugurals. And while the military routinely evaluates service members for radical connections, the FBI screening is in addition to any previous monitoring.
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Multiple officials said the process began as the first Guard troops began deploying to D.C. more than a week ago. And they said it is slated to be complete by Wednesday. Several officials discussed military preparation on condition of anonymity.
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