Did Dr. Sloper have an ulterior motive in Washington Square?
It is thoroughly established that Catherine already has an income of $10,000 a year inherited from her mother. Dr. Sloper has control of this income, and he is a man who cares a lot about money. Is it possible that his objection to Catherine's marrying Morris Townsend is partly due to the fact that he would be losing control of her substantial income? If $10,000 represents interest, then the capital sum must be around $250,000, which would be equal to around a million in purchasing power in 1840.
Dr. Sloper has plenty of faults and flaws, but stinginess is not among them. If he were close with his money to the extent that you are insinuating here, would he allow his sister to live with him for decades?
The opposition that Sloper puts up against Morris Townsend seems, at first, a completely honest and paternal opposition. This opposition morphs into stubbornness and a will to control his daughter.
While Sloper does not like his daughter, Catherine, he nonetheless engages in a battle of sorts with her fiancee. He refuses to give up his daughter to the man though Morris is like-able enough. Sloper repeatedly says that Morris is not, however, like-able as a son-in-law. The man is out to take advantage of Catherine. This becomes established fact as the novel goes on.
The doctor chooses to stand by his observation, staunchly and stubbornly. He wants to be right. In this regard he is not generous at all but is, rather, extremely stingy. When he is right, he needs everyone to know it, to acknowledge it, and to acquiesce.
This is a well-demonstrated part of his character.
Deeper complexities may exist. Sloper may be over-protective of Catherine because he is guilty for not liking her. Sloper may be taking revenge against Catherine for not being her mother - the only person who Sloper ever truly loved. Sloper may be perversely enjoying his daughter's drama to such an extent that he, like Mrs. Penniman, cannot let it go even when Catherine can.
Sloper may be childish and petty at heart, however observant and brilliant besides.
Evidence seems to be lacking that would show him to be interested in maintaining control of Catherine's fortune for himself. He does not spend money lavishly. There are no descriptions of either need or use for Catherine's income.