To begin with Emma, herself, does not truly believe the arguments she makes to Mr. Knightley on behalf of Frank. Earlier when speaking to Mrs. Weston Emma says: "He (Frank) ought to come," said Emma. "If he could stay only a couple of days, he ought to come; and one can hardly conceive a young man's not having it in his power to do as much as that. A young woman, if she fall into bad hands, may be teazed, and kept at a distance from those she wants to be with; but one cannot comprehend a young man's being under such restraint, as not be to able to spend a week with his father, if he likes it."

On mentioning to Mr. Knightley yet another delay by Frank Emma "proceeded to say a good deal more than she felt,..." and "found herself directly involved in a disagreement with Mr. Knightley; and, to her great amusement, perceived that she was taking the other side of the question from her real opinion, and making use of Mrs. Weston's arguments against herself."

At this point in time, visiting was a duty, whether it was someone down the street, across town or sixteen miles from London. Mr. Knightley says: "There is one thing, Emma, which a man can always do, if he chuses, and that is, his duty;..."

It doesn't help that the Westons are receiving letters from Frank that he goes other places when he wishes. This, in my opinion, makes it inexcusable that Frank has not shown up. It is doubly an insult not to visit his father, because he has acquired a new wife. Mrs. Weston's feelings are obviously hurt on top of it all.

From Jane Austen's Guide to Good Manners: "'Everybody who comes to Southampton finds it either their duty or pleasure to call upon us', the Authoress recorded ironically in 1804. Calls which may be considered a duty rather than a pleasure are those 'visits in form' which must be paid to the newly-married; the bereaved; and those in straitened circumstances, who are all too liable to feel themselves neglected. Not to wait upon a bride is very remiss... This is a matter of mere common politeness and good breeding frets old Mr. Woodhouse in EMMA. He is correct."

As far as Mr. Knightley's reaction is concerned, he is a man of honor. He does not suffer gladly a lapse in ill-manners, poor ethics or meddling where you shouldn't and it seems to me, he levies his opinion/anger equally whether at Emma, Frank or whoever, at perceived injustices. As ever, my humble opinion.