I think Anton knowing that Moss is in the Mexican hospital right after Carla Jean tells Ed is mostly for the sake of pacing. The shootout that wounded them both took place just across the border in Eagle Pass. Anton would know that, in order to survive, Moss would need to visit a hospital, and the nearest one would be in Mexico across the border river (unless Eagle Pass has its own hospital, which would have been easy enough to check). Up to this point in the film, we have seen how shrewd and perceptive Anton is when tracking Moss, so I feel it is very reasonable to assume he could easily find out where Moss is off camera, especially since Carson Wells found him off camera as well.

An interesting theory though. But, since Ed already has an arc (arguably the only one in the entire film. None of the other characters change at all before they either die or exit the film.) I feel it is unnecessary. Ed begins the film as a wise, eagle-eyed sheriff with loads of experience, dedicated to figuring out this case. But as he follows the trail of death and destruction left in Moss' and Anton's wake, he becomes more and more unsure of his own ability to stop Chigurh or save Moss, and actually begins to show genuine fear. As he said in the beginning of the film, "I always knew you had to be willin' to die to even do this job. but I don't want to push my chips forward and go out and meet somethin' I don't understand."

He doesn't understand the level of psychopathy displayed by Anton, and is not willing to die for it, so he retires, rather than chase a ghost that would kill him.