It is very unlikely that the public would support mandatory voting in the US and in Texas in particular. There are at least two reasons why this is likely to be the case.
First, we should look at the idea of personal rights. Americans guard their personal rights rather jealously. This is particularly true in conservative states such as Texas. For example, in Texas, people over 21 years of age may ride motorcycles without wearing helmets. Texas took this step very soon after the federal government allowed states to lift the helmet requirement. States without such laws typically argue that Americans should be able to behave as they choose. If Texans are not willing to have a mandatory helmet law when such a law would help save lives, there is little reason to think that they would welcome the government telling them they had to vote.
Second, we should acknowledge that many people in Texas would, for partisan political reasons, not want more people to vote. Texas is a Republican state. In general, Republicans and conservative voters are more likely to vote than liberals and Democrats. Mandatory voting would force more poor voters and minority voters to turn out. This would not help Republicans. This makes it even less likely that Texas would support such a law.
Thus, mandatory voting is not ever likely to happen in the US and Texas is not a good candidate to take the lead in making voting mandatory.