In a statement, Abbott extended his "deepest condolences" to Scalia's family.
The San Antonio News-Express said Scalia had apparently died of natural causes while visiting a luxury resort in West Texas.
Scalia, appointed to the top U.S. court in 1986 by President Ronald Reagan, was known for his strident conservative views and theatrical flair in the courtroom.
His death comes as the court is set to hear its first major abortion case in nearly 10 years, and ahead of key cases on voting rights, affirmative action and immigration.
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