Most of the crimes Columbo "solves" wouldn't hold water in a court of law with a great defense attorney, like Matlock.

I agree, the episode would've worked better had Mayfield overheard her suspicions rather than being confronted by them. Also, Mayfield took a big chance by killing her in the parking lot of a hospital when people are coming and going at all hours. It would've played better had he killed her in her apartment where he was less likely to have been seen.
Jesus is the Son of God and my Lord and savior. Pretty cool.

Fortunately Nurse Martin suspected something was wrong with the sutures which would've lead to an investigation. Unfortunately, she told the person she suspected of what she thought.
Which opens up another hole in this Swiss cheese of a plot. Not impossible she might have gone to him first, but reallyif you actually suspect someone of capability of cold-blooded murder, are you all that apt to go to them and tell them of your strong suspicion that they've committed one? (Especially when the party in question has as much to lose as Mayfield did.)
Again, the performances in this one are the episode's chief strengththe story itself is largely flimsy and ridiculous in its hinging on contrivances.
50 Is The New Cutoff Age.

Dr. Mayfield probably felt his chances of getting away with murder were pretty good. If a person dies under the care of a doctor and in this case Mayfield was Hidemann's doctor, and the death was related to why he was under a doctor's care (his heart condition,) then chances are that an autopsy would've been ruled as unnecessary due to lack of suspicious circumstances surrounding the death. Fortunately Nurse Martin suspected something was wrong with the sutures which would've lead to an investigation. Unfortunately, she told the person she suspected of what she thought. The really odd thing about this episode was that Mayfield (clearly a sociopath) ended up being arrested for two murders; neither of which was his intial target.
Jesus is the Son of God and my Lord and savior. Pretty cool.

I don't know what the statistical percentage was of autopsies performed in the state of California, during the 1970's, on patients who died on the operating table; however I'm betting it was still high enough to make this a large and problematic plot hole for this episode.
According to the CDC, the incidence of autopsy performed on death from disease, throughout the US, was 79% in 1972, as opposed to 46% in 2007, making it
more
likely that the victim's death would have been investigated by the procedure at that time, not less.
http://www.webmd.com/healthy-aging/news/20110804/autopsy-rate-shows-big-decline
As careful as Mayfield was of his reputation, it would seem pretty unlikely that he'd have taken even the minor risk of his crime being discovered in that manner. Being a doctor, he certainly must have had some idea that his victim undergoing a post-mortem examination was not all that remote of a possibility. I like Nimoy's performance in this episode, as well as his and Falk's testy interplay, but plot-wise, I still rank this episode as one of the weakest in terms of overall probability.
50 Is The New Cutoff Age.

"Supposing Autopsy"
an ailing person died on the Operating Table
even today unless some one gets a wild hair; an Autopsy is maybe only a 50% chance in the 70s it'd have been way less
it's the old Horse v Zebra argument

I've never seen how the dissolving suture scheme would have worked anyway. Supposing Heidermann had died and been autopsied? An autopsy would have revealed that incorrect thread had been used for the sutures, thus opening up Mayfield, at the very least, to a charge of gross negligence or incompetence even if not deliberate murder. How would his professional reputation have survived that?
50 Is The New Cutoff Age.

Hide it in the body? The other doctors and nurses assisting would see it.

Not even close.
Columbo finds some thread in his pocket. He can't prove that Mayfield put it there. He can't even prove it came from the guy's chest. And he has no evidence at all that he killed the nurse.
I always wondered why Mayfield didn't just tuck the thread into the body when nobody was looking and leave it there when he closed the guy up. Nobody could search for it there, and it would dissolve away within another couple of days. No need to smuggle it out of the operating room at all.

This discussion reminds me of a spoof that Mad Magazine did on the forced retirement of various TV characters, including Columbo. Columbo is being forced to retire, and he protests, citing his great arrest record. But then the chief points out that all of them were arrested on evidence that didn't hold up in court. Then they were acquitted, and sued the department for false arrest. The chief says they can't afford it anymore. So Columbo goes home, and gets shot by his wife when she realizes that he will be home, talking, 24/7!

Best answer is likely that it's a TV show not reality. I like the show but in most episodes any criminal as smart as they were supposed to be in this show would have lawyered up immediately. Not good for tv show but in reality that's what happens. And with all evidence mishandling, search & seizure without warrants even a public defender could file a motion to suppress that would likely be granted forcing the DA to dismiss the case. Columbo wouldn't be a real cop for very long with his methods.

I don't see how Columbo established any means or opportunity. Sharon Martin was killed in a publicly accessible parking garage. Anyone could have done it. How did he establish that Dr. Mayfield was the only one with the means and opportunity?
Motive was the only one he could establish. In the courtroom, a capable defense lawyer can easily write it off if there is no evidence of committing the actual act.

Means not method.

What he
could
have done, though, is to bring an extra suture with him and leave it somewhere where the police would find it, so that they would stop looking. An extra suture which he would hide slightly less well than the real one. Damn,
I
should've been the killer.

I suppose that Columbo could always call in Dr. Quincy to test the suture for Dr. Hidemann's blood.
Jesus is the Son of God and my Lord and savior. Pretty cool.

It is still all circumstantial, however. And the dissolvable suture from the OR would almost certainly be inadmissible as tainted evidence, given that it was found in Columbo's pocket and its providence could therefore not be verified. And given Columbo's history of rather careless handling of evidence, and his dogged determination in targeting Nimoy's character (even losing his temper at one point), it would not be too difficult for the defence to argue that Columbo was an untrustworthy witness, and the discovery of the suture in his own pocket as suspect.

The usual steps in linking a suspect to a murder are establishing motive, method and opportuity. I think Columbo was able to establish all three. Since the toughest one to prove was motive, the other two would be easier to prove. Link all of the circumstantial evidience together including the death of Harry Alexander and testimony from her room mate Marcia, and I don't think a DA would have a hard time getting a murder conviction.
Jesus is the Son of God and my Lord and savior. Pretty cool.