Bishop Bargallo, who is also president of Caritas Latin America, appeared on the channel to read a statement in which he denied having any inappropriate relationship and described the woman as a "childhood friend."
He also acknowledged, "It has caused a bad impression."
"I ask forgiveness if this has caused some sort of damage," he said. "I'm totally committed to God, the church and serving the diocese."
The source of the photos has not been explained, although the A24 website mentioned the bishop has had tense relations with officials in the Argentine government over social policy.
Why Bishop Bargallo was in Mexico or how he paid for a stay at a luxury resort also was not been explained.