Dr. Sloper is strongly opposed to his daughter marrying Morris Townsend. She tells her father, however, that she is engaged and intends to marry Townsend. The best her father can hope for is to delay the marriage. He thinks there is some possibility that Catherine will get over her infatuation and forget about Townsend as a result of being distracted and changed by all the fascinating sights and social activities offered by Europe. The doctor will also have her to himself and perhaps be able to get her to see Townsend for what the doctor knows him to be, a fortune hunter and a parasite. There is, of course, the further possibility that the attractive Morris Townsend will find somebody else who has more to offer in looks and personality, and nearly as much to offer in financial expectations. Catherine shows little interest in Europe. She has complete faith in Townsend. She thinks about him all the time and corresponds with him in secret. Morris takes advantage of Dr. Sloper's absence to make himself at home in the mansion on Washington Square and to cultivate the affection and devotion of Lavinia Penniman. Dr. Sloper cuts the Europe trip short when he realizes that Catherine is not going to give Morris Townsend up. She tells her father she intends to marry Townsend as soon as they get back to America. 

According to the summary and analysis in the enotes study guide:

Catherine comes back changed, but not as her father had envisioned. She is stronger and more independent, less reliant on earning the approval of her father. She has discovered that her father is not worth the compromise that she must make to earn his love. Not only has she given up hope that he will relent in the matter of her marriage to Morris Townsend, but she vows never to take anything from him again. 

When Townsend realizes that he is never going to get his hands on Dr. Sloper's money, he decides to jilt Catherine. He is willing to sell himself for $30,000 a year but not for her personal income of $10,000 a year. This comes out clearly in a conversation he has with Aunt Penniman in Chapter 28:

Morris walked along a moment, and then he repeated harshly, "I must give her up!"

"I think I understand you," said Mrs. Penniman gently.

"I certainly say it distinctly enough--brutally and vulgarly enough."

He was ashamed of himself, and his shame was uncomfortable; and as he was extremely intolerant of discomfort, he felt vicious and cruel. He wanted to abuse somebody, and he began, cautiously--for he was always cautious--with himself.

"Couldn't you take her down a little?" he asked.

"Take her down?"

"Prepare her--try and ease me off."

Mrs. Penniman stopped, looking at him very solemnly.

"My poor Morris, do you know how much she loves you?"

The trip to Europe serves mainly to show how strongly Catherine loves Morris Townsend and to suggest how totally devastated she will be when he goes away on a fictitious business trip to New Orleans and makes it clear that he is not coming back.