Would the Texas Supreme Court become the highest court in the land?

Yes. In an independent Texas, no court outside Texas has any authority over Texas law, so the top of the Texas judiciary becomes the final word. Texas is unusual in having two high courts, and both would sit at the summit of an independent Texas legal system.

No appeal would run to Washington

Today a narrow set of Texas cases can be appealed up to the U.S. Supreme Court on federal questions. Independence ends that. Once Texas is its own nation, there is no higher court in Washington and no federal review of Texas decisions, because Texas no longer answers to the U.S. Constitution. The Texas judiciary becomes self-contained, with its final appellate courts as the last stop. That is what judicial sovereignty means: Texas cases are decided in Texas, all the way to the end.

Texas has two high courts, not one

Here is a detail many people do not know. Unlike most jurisdictions, Texas splits its highest judicial authority between two courts. The Supreme Court of Texas is the court of last resort for civil and juvenile matters. The Texas Court of Criminal Appeals is the court of last resort for criminal matters. This two-court structure already exists and already functions as the top of the Texas system for everything except the slice of federal questions that currently goes to Washington. Remove that slice at independence and these two courts simply become the final word on everything, with nothing above them.

This is a promotion of authority, not a new institution

An independent Texas does not have to build a supreme court. It already has the courts, the justices, the procedures, and the body of Texas law they apply. What changes is the ceiling. The Supreme Court of Texas and the Court of Criminal Appeals stop being subordinate on federal questions and become fully sovereign. The institutions are the same the day after independence as the day before. Their authority is simply complete.

The law they apply is already Texas law

These courts already interpret the Texas Constitution and Texas statutes every day, and on questions of Texas law they are already the final authority. The Texas Constitution has been the governing charter of this state for 150 years, with its own Bill of Rights that in several respects protects Texans more strongly than the federal one. An independent Texas is not asking its high courts to apply some untested new code. It is asking them to keep doing what they already do, now without a federal court able to override them.

The bottom line

Yes. With no appeal running to Washington, the Supreme Court of Texas and the Court of Criminal Appeals become the final courts of an independent Texas, civil and criminal. They already sit atop the system and already apply Texas law. Independence removes the only authority that ever sat above them.