I am not sure which action you are referring to, so I am assuming it is his decision to run away and then subsequently killing himself. It has been a great debate about whether Paul really knew what he was doing or he simply did not realize the seriousness of what he did. He was driven by the desire to live the life he always wanted to live, which he could not do where he was, so he ran away to New York City after stealing from his employer. Obviously, stealing was a serious offense, but running off to live his dream was not, in my opinion. As for whether Paul regretted stealing and running off to New York City, Cather leaves the reader to ponder this question. We do not have a clear answer. According to eNotes:
Cather's characterization of Paul is ambivalent, and readers are left to wonder whether Paul freely chose his tragic end or not. While Paul's alienation from his environment is clear, the reader cannot tell whether Paul's is a "case" of environmental determinism or of the folly of youth, of a dreamer who died with "all his lessons unlearned."
It depends on how you're defining the end. The next to the last paragraph includes this line: "As he fell, the folly of his haste occurred to him with merciless clearness, the vastness of what he had left undone. There flashed through his brain, clearer than ever before, the blue of Adriatic water, the yellow of Algerian sands."
That's regret, pure and simple. To think of folly and things left undone is regret. However, look at the very last line of the story: "Then, because the picture-making mechanism was crushed, the disturbing visions flashed into black, and Paul dropped back into the immense design of things."
I'd say he regrets, and then, as he dies, so do his regrets.
If Paul feels regret, it is certainly as superficial as his other feelings throughout the story. I do not believe that Paul had some meaningful epiphany that he was throwing his life away- he had hours beforehand to realize that. Perhaps Paul realized that he had not played his part as well as he thought. He was trying to live as he imagined a young, wealthy gentleman would and to enjoy the luxury and glamour of upperclass life. Maybe right before the end, he realized that he hadn't had the full experience of wealth yet- he hadn't traveled.
That's just my view, and there is no substantial evidence in the text to support it, besides illustrations throughout the story of Paul's "case".