 Original Message

 > Is there a simple proof of Fermat's principle?

 Feynman, volume II chapter 19.

 Probably as simple as it's going to get.
 Recommended. Really, really, recommended.
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In the context of say "Mechanics" then what Feynman says can be said to
prove Fermat's principle from Newton's laws and their generalization to
energy concepts (KE and PE). So I think this is an excellent reply to the
question of "Is there a simple proof of Fermat's principle for mechanics
starting from Newton's laws. Or even the question as originally stated.
Similar statements could be said in the context Fermat's principle in the
context of quantum mechanics, optics etc.
However, in another sense one could reply to the question by stating that
one can view Fermat's principle as fundamental organizing principle and
hence it requires not proof other than how well it serves as an organizing
principle for the data.
In the context of mechanics this is how one might respond to the question of
"Is there a simple proof of Newton's 2nd law?" typically, assuming one
wants to use it as a central organizing principle of mechanics, one states
it without proof and appeals to experiment (data) for its validity.
Of course, one could say, yes, starting with Fermat's principle for
mechanics one can easily derive Newton's 2nd law.
Subnote: I would argue that Fermat's principle (I really mean least action
principles) serve as a broader organizing principle in Physics and are
therefore probably preferable as such, than say Newton's laws.