TexasCHLforum.com

Dicion -- I believe you have to read the entire exception in Section 30.06:

(e) It is an exception to the application of this section
that the property on which the license holder carries a handgun is
owned or leased by a governmental entity and is not a premises or
other place on which the license holder is prohibited from carrying
the handgun under Section 46.03 or 46.035.

Note that for this exception to apply it has to be government owned or leased AND not listed as a prohibited place under 46.03 or 46.035. Hospitals are listed under 46.035 as prohibited places, as long as they give effective notice. Public libraries, DPS license offices, the Tax Assessor, etc are not listed as prohibited places, so the exception in 30.06 applies to them.

A hospital is one of the locations specified in law that may ban CHL carry, as long as the hospital "provides notice" as outlined in 30.06. This appears to be true regardless of whether it is a public or private hospital. Altho there is a section of the law that generally says that CHLs can't be banned from public property, there are several exceptions to this, e.g. correctional facilities, secured areas of police stations, meetings of governmental entities, school buildings and school premises in certain circumstances, courtrooms....

Also, as far as a private entity leasing public property, it would appears that the private entity may implement whatever rules it likes, regardless of whether the property is owned by the state or local government. It might appear otherwise in state law, but in practice this is the case. The American Airlines Center in Dallas being a prime example.