The senator said it was time for the Senate to abolish the filibuster in order to guarantee abortion rights.
"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.
GoDaddy informed Texas Right to Life, the group behind the website, that it had violated "multiple provisions" of its terms of service.
The criticism from Bishop Donald Hying comes after the president, a Catholic, said that he doesn't believe that life begins at the moment of conception.
The clinic has been forced to limit patient numbers for the first time in its history.
The former Lone Star state Democratic congressman told supporters the new strict abortion law was part of a "terrible trajectory."
"I suggest that all women refuse to have sex with men until they are guaranteed the right to choose by Congress," the singer and actress tweeted.
Public figures have urged people to sign a Planned Parenthood petition against the law which effectively bans abortion after six weeks of pregnancy.
Dan Patrick told Fox News that he was "proud" that Texas passed Senate Bill 8 which the Supreme Court refused to block.
"While we fight back, know you still have options. Pills can be mailed to your home discreetly," Representative Ocasio-Cortez tweeted.
The anti-abortion website violated the web hosting terms of service because it asked people to share strangers' personal medical details.
The Oklahoma law stipulates that any doctor who performs an abortion after a fetal heartbeat is detected would be committing homicide.
To do away with the filibuster, it would require the vote of all 50 Democratic Senators, plus a tie-breaking vote from Vice President Kamala Harris.
Republicans across the country have expressed a desire to also try and pass measures mirroring the new Texas law making abortions almost entirely illegal.
Some Democrats were already arguing for an expansion of the court before the controversial decision on Wednesday.
President Biden is wrong: The Supreme Court lacks the "supreme authority to ensure justice could be fairly sought."
"Elon consistently tells me that he likes the social policies in the state of Texas," Texas Governor Greg Abbot said.
The Supreme Court voted 5-4 in the middle of the night to deny an emergency appeal from abortion providers aimed at blocking the Texas law from taking effect.
Alex Arnold said: "Pro-life is the most grotesque misnomer. It is forced birth," as she also called for complete abortion access.
One of the Lone Star State's newly implemented laws bans the discussion of critical race theory in classrooms.
Texas has introduced a law banning abortions after six weeks of pregnancy, which could be used against anyone who drives a woman to a clinic or giving financial assistance for the procedure.
The former presidential candidate warned in February 2016 that the Supreme Court could roll back abortion access.
The New York Democrat criticized what she called "far-right minority rule" following the 5-4 ruling.
Private citizens are now able to sue an individual who "aids or abets" a woman in getting an illegal abortion.
The new Texas law bans providers from terminating pregnancies once a fetal heartbeat is detected.
The court's order said the decision was "not based on any conclusion about the constitutionality" of the law.
"The court unilaterally declared abortion 'a fundamental right' under the Iowa Constitution, undermining the state's immense interest in protecting children and mothers," said Alliance Defending Freedom Legal Counsel Elissa Graves.