For the Purpose of laws in Texas, it does not matter who owns the land. If it has a historical marker it is under the protection of the state. I could own an old house, and register it with the state, and have a plaque put on it. Once that is done even though I owned the land and house, it is a protected site and I would be breaking the law to metal detect on it.
Texas also has the antiquies law that pretty much covers everything, even on private property. If it is 100+ years old it is a NO NO to dig or remove it.
So the answer is NO you would be breaking state law if you detected it. Now that being said, if you are talking about the oblate church off of military highway in the RGV, that place has been detected more times than naught, and yes once someone was caught and ran off from there people tend to keep an eye on the place now because they are trying to restore it and it was vandalized a couple of times. However out of the 8+ people I know who have detected it, not one of them found a thing. Just because the others detected it and did not get caught still does not make it legal. Here is what the law says:
Natural Resources Code
191.092 Other Sites, Artifacts, or Articles.
Sites, objects, buildings, artifacts, implements, and locations of historical, archeological, scientific, or educational interest, including those pertaining to prehistoric and historical American Indians or aboriginal campsites, dwellings, and habitation sites, their artifacts and implements of culture, as well as archeological sites of every character that are located in, on, or under the surface of any land belonging to the State of Texas or to any county, city, or political subdivision of the state are state archeological landmarks and are eligible for designation.
191.093 Prerequisites to Taking, Altering, Damaging, Destroying, Salvaging, or Excavating Certain Landmarks.
Landmarks under Section 191.092 of this code are the sole property of the State of Texas and may not be taken, altered, damaged, destroyed, salvaged, or excavated without a contract with or permit from the Antiquities Committee.
191.131 Contract or Permit Requirement.
(a) No person, firm, or corporation may conduct a salvage or recovery operation without first obtaining a contract.
(b) No person, firm or corporation may conduct an operation on any landmark without first obtaining a permit and having the permit in his or its possession at the site of the operation, or conduct the operation in violation of the provisions of the permit.
191.132 Damage or Destruction.
(a) No person may intentionally and knowingly deface American Indian or aboriginal paintings, hieroglyphics, or other marks or carvings on rock or elsewhere that pertain to early American Indian or aboriginal habitation of the country.
(b) A person who is not the owner shall not willfully injure, disfigure, remove, or destroy a historical structure, monument, marker, medallion, or artifact without lawful authority.
191.133 Entry Without Consent.
No person who is not the owner, and does not have the consent of the owner, proprietor, lessee, or person in charge, may enter or attempt to enter on the enclosed land of another and intentionally injure, disfigure, remove, excavate, damage, take, dig into, or destroy any historical structure, monument, marker, medallion, or artifact, or any prehistoric or historic archeological site, American Indian or aboriginal campsite, artifact, burial, ruin, or other archeological remains located in, on, or under any private land within the State of Texas.
191.171 Criminal Penalty.
(a) A person violating any of the provisions of this chapter is guilty of a misdemeanor, and on conviction shall be punished by a fine of not less than $50 and not more than $1,000, by confinement in jail for not more than 30 days, or by both.
(b) Each day of continued violation of any provision of this chapter constitutes a separate offense for which the offender may be punished.