Does Texas require general contractors to carry liability insurance?

No, the state of Texas does not require it as of the time of this answer.

However, most Texas clients are financially savvy and are not likely to offer a contract or even accept a bid from an un-insured contractor.

The first hallmark of a reputable contractor is that they come properly insured for the services being offered by contract bid.

In other Words, Your bid proposal should include documentation signifying that you are insured and that your coverage meets any minimum requirements of the bid as proffered.

If your an un-insured contractor, you can't expect serious consideration of your bid. You also might not be invited to bid for that client again in the future. Proper insurance and required coverage limits are generally part of the bid requirement to begin with.