You nailed it. Dr. Flemming did say Friday at 2. However, I don't think there is a single clue or reason as to why he started focusing on her and conducted the interrogation, other than it's part of the detective process and the episode's progression with its cumulative "little things", e.g. the exchange outside the hospital door following Mrs. Flemming's passing. That said, the discussion/ showdown between Flemming and Columbo in the doctor's office is at the heart of this movie and very much puts Columbo on a mission, and he quickly zeroes in on the weak link.
I watched the episode again. At 34 minutes and 29 seconds into the episode, before Columbo and Dr. Fleming go to the hospital, Joan Hudson indeed calls. Dr. Fleming makes it seem to Columbo like the caller is a patient. Thanks to your hints, guys, I now see that a patient calling Dr. Fleming's home number is indeed suspicious. And I now also connect Columbo's deliberate visit at 2 o'clock on Friday, to find out the name of that home-caller. Thanks! I can even add: Earlier, Dr. Fleming explicitly asked Joan not to call/visit Dr. Fleming for a few days. When Joan does call Dr. Fleming and apologizes for calling, Joan actually apologizes for ultimately exposing Dr. Fleming as the murderer.