This man was not committed to death row because of one doctor. He was found guilty because multiple people in his life testified he had a history of violently shaking and screaming at a child for crying.
IANAL, but wouldn't that be against the rule that character evidence cannot be used by the prosecution (unless in countering character evidence from the defence)?
Specifically, I believe what you describe would be in contravention of the following: https://www.law.cornell.edu/rules/fre/rule_404
> (a) Character Evidence.
> (1) Prohibited Uses. Evidence of a person’s character or character trait is not admissible to prove that on a particular occasion the person acted in accordance with the character or trait.
> [...]
> b) Other Crimes, Wrongs, or Acts.
> (1) Prohibited Uses. Evidence of any other crime, wrong, or act is not admissible to prove a person’s character in order to show that on a particular occasion the person acted in accordance with the character.
> (2) Permitted Uses. This evidence may be admissible for another purpose, such as proving motive, opportunity, intent, preparation, plan, knowledge, identity, absence of mistake, or lack of accident.