"If I am beaten and raped in the state of Texas, I have to give birth to the baby of my rapist," the narrator says in the video's opening.
Republican state Representative Briscoe Cain said the amendment was to make sure that "people that do innocent things" are not arrested again for their past actions.
At least 50 Texas Democrats departed the state for Washington, D.C., in protest of restrictive new voting laws proposed by Texas Gov. Greg Abbott and their other GOP counterparts.
While O'Rourke has yet to confirm a 2020 run, he has explored the possibility in recent weeks.
In an analysis of early voting numbers, Target Smart found that more than 200,000 people have already voted for the first time in Texas, and the number of Hispanic voters has increased by more than 200 percent since 2014.
A Quinnipiac University Poll found that Cruz was leading O'Rourke by five points, with 3 percent of Texan voters still undecided.
"There is a loss of civility, there is an anger, there is a rage on the far left that is frightening," the senator said. "The images of the left-wing mob beating on the doors of the Supreme Court—that's not good for the country."
Polls last month had offered a major boost to Beto O'Rourke's hopes, but that was before the Kavanaugh Supreme Court nomination fight.
"I am so grateful to him for his service; he's going to go down as one of the greatest presidents. And yet, this [election] is on Texas," said O'Rourke.
"I'm, like, Wow, that's a pretty extreme position, if you're Bill de Blasio," Trump Jr. said, referring to New York City's mayor.
The race has become one of the most closely watched in the country, with polls putting O'Rourke within a margin of error of Cruz.
Beto O'Rourke is narrowing the gap between him and Texas Senate opponent Ted Cruz, particularly among independents.
Democrats are eying a first senator from Texas in three decades.