McMurtry draws the characters of Gus and Woodrow as polar opposites. Gus, is warm, charming, and adept at interpersonal relationships. Woodrow is cold, strait-laced and emotionally distant. It's Woodrow Call's character that keeps him from recognizing Newt as his son.
We see Call as unable to relate to women. He avoids contact with them and they with him. Clara and Lorie show how poorly Call interacts with women; Clara is very vocal in her dislike of Woodrow and Lorie avoids him. This inability to interact with women explains why Call's only relationship with a woman was with Maggie. In her professional capacity, he does not have to emotionally engage with her. Consequently, when she has his child, he is unable to make an emotional attachment to either of them.
Duty, pride and honor also play a part in Call's public denial of Newt. Call has a strict set of rules for behavior. We see this when he honors his promise to Gus to bury Gus in Texas, no matter how difficult the trip. In Call's mind, his visits to Maggie displayed a weakness in him and his son by a prostitute was unacceptable to his code of conduct. He repeatedly references Maggie's profession as an excuse to deny Newt his name.
Call does recognize Newt in other ways. We know that Gus and Call raise Newt after Maggie dies, not a small step for two Rangers with no experience raising small children. Woodrow also reacts violently when Newt is attacked when the Army tries to requisition Newt's horse. Call's blind rage is much greater than it would have been if it had been Deet's horse or Pea Eye's. The most profound recognition, however, comes when Call gives Newt the "Hell Bitch." If ever there was a moment to acknowledge Newt, that was it. But Woodrow fails once again and leaves without acknowledging Newt.