Is a son considered disobedient to his father if he asks him not to bring his second wife into his private home, and the father quarrels with him and refuses to reconcile?

The father should not be prevented from entering the house merely because he is accompanied by his other wife; rather, his wife should be honored out of deference to him. If you feared for your mother's feelings, it would have been possible to conceal the matter from her. Since you have apologized to your father, he should accept the apology, and you should strive to appease him. If that does not work, then there is no blame upon you, and you are not considered undutiful to him.