The bill, named HB20, bans social-media companies with over 50 million monthly active users from censoring people because of "the viewpoint of the user or another person."
The bill enables private citizens and the Texas attorney general to sue tech companies that they think have unjustly kicked someone off a platform. Abbott is among some Republicans who insist Big Tech companies unfairly target conservatives with their moderation rules.
"There is a dangerous movement by some social media companies to silence conservative ideas and values," Abbott announced in a video posted on Twitter.
This comes four months after Florida Gov. Ron DeSantis signed a similar bill trying to stop "deplatforming" on social media. A federal judge barred his bill, ruling it broke companies' First Amendment rights to get rid of content that violated their standards.
The Texas bill is due to take effect on December 2, and digital rights group NetChoice — which brought a legal case against DeSantis' Florida bill — announced in a statement the bill was "unconstitutional and dangerous."
"HB20 has the same First Amendment flaws as the Florida law that a federal court blocked this summer. The same outcome will almost certainly occur in Texas," NetChoice president Steve DelBianco said in a statement.
"Moderation of user posts is crucial to keeping the internet safe for Texas families, but this bill would put the Texas government in charge of content policies."
All this is taking place in the wake of the ban of Former US President Donald Trump from Facebook and Twitter after a group of his followers attacked the Capitol in January.
The social networks have all rejected stifling conservative viewpoints.
Yet, they do enforce terms of service which prohibit content like incitement to violence and co-ordinated disinformation.
"Social media websites have become our modern-day public square," announced Texas governor Greg Abbott, after signing the bill into law on Thursday.
"They are a place for healthy public debate where information should be able to flow freely.
"But there is a dangerous movement by social media companies to silence conservative viewpoints and ideas."
Furthermore, in May, Florida passed a law that banned social networks from de-platforming politicians.
Though, some parts of that bill were suspended by a federal judge, who ruled that it violated the First Amendment right to free speech.
Another Texas law, changing the rules around abortion in the state, is currently being challenged by the US Department of Justice.