The short answer is no. Tom Ripley is not capable of feeling those emotions because he is a sociopath. He always feels justified in acting solely to further his own best interests.
In the early stages of the novel, Tom seems to be an inefficient petty criminal, but he never shows any sign of regretting the fraud and thefts.
After Tom gets involved with Dickie, he does not set out to get rid of him and assume his identity. Rather, he uses his sharp survival skills to play it by ear and figure out what he might gain at each step of the way. His violent outburst and brutal manner of killing Dickie generates some problems, in relation to the blood and the boat, but he does not regret his actions.
In regard to Freddie, Tom sees his path as clearly defined. He can see in Freddie no redeeming qualities. Dickie was at least charming and attractive, but Freddie is a boor. He presents only a danger because he does not buy Tom's story about the reasons for being in the house. There seems no safe, logical way to deal with Freddie and prevent him from sharing his suspicions. Possible feelings of regret, but not remorse, stem from the fact that killing Freddie means that Tom must now resume being himself and turn Dickie into the suspected murderer.