Would Texas degrees still be recognized in the U.S. and abroad?

Yes. Degrees are recognized through accreditation and reputation, not through which government a university sits under, and Texas universities are accredited and respected worldwide today. Nothing about independence undoes a diploma already earned or a university's standing.

Recognition does not come from Washington in the first place

Here is the fact that dissolves the worry. The U.S. federal government does not certify or validate college degrees. There is no central federal body that stamps a degree as legitimate. In the United States, recognition rests with the receiving party: the university you transfer to, the employer who hires you, or the state licensing board that licenses your profession. So a Texas degree was never "recognized" by Washington to begin with. Independence cannot take away a recognition the federal government does not grant.

Accreditation is private, not federal, and Texas universities already have it

Degrees carry weight because the institution behind them is accredited, and accreditation in the United States is done by private, nongovernmental organizations, not by the government. The U.S. Department of Education only recognizes those private accreditors; it does not accredit a single school itself. Texas universities are accredited today by these independent bodies, and that accreditation does not evaporate because Texas becomes a country. The standing of the University of Texas, Texas A&M, Rice, and the rest rests on their academic quality and their accreditation, both of which Texas keeps. (The next answer covers how Texas handles accreditation going forward.)

Degrees are recognized worldwide on reputation, which Texas keeps

Internationally, recognition works the same way: there is no single global authority, and a degree's worth abroad depends on the university's reputation and on credential evaluation by the receiving country's own bodies. Texas universities are already known and respected around the world, enrolling international students and placing graduates globally. A degree from a strong Texas university is recognized abroad because the university is excellent, not because of the flag over its campus. Graduates of universities in Canada, Britain, Australia, and every other country have their degrees recognized internationally, and Texas graduates would be in exactly that company.

Already-earned degrees are settled and permanent

For anyone who already holds a Texas degree, there is nothing to worry about at all. A diploma already conferred is a completed fact. It does not get reissued, revalidated, or downgraded because the issuing state becomes a nation. The millions of Texas degrees already in the world stay exactly as valid as the day they were earned.

The bottom line

Texas degrees stay recognized because recognition flows from accreditation and reputation, not from Washington, and Texas universities keep both. Degrees already earned are permanent, and Texas graduates are recognized at home and abroad on the strength of their institutions, which independence leaves intact.