I'm not totally sure; I think it depends on what exactly you classify as calculus of variations, and what assumptions you're willing to make along the way. For what it's worth, I don't think I know how to prove the most general case possible. But given that I encountered this in physics rather than in math, I was satisfied with proving the physical analog, which would be the claim that the path of light in Euclidean space is always linear. Assuming we're talking about light allowed me to make 3 useful assumptions: (1) that the speed of the particle doing the traveling is constant, which comes in quite handy in the proof [1], (2) that there is only 1 independent variable (i.e., time) rather than 3 space coordinates, and (3) that our functions are all sufficiently smooth and such. The nice thing about this proof is it doesn't need the Euler-Lagrange equation that everybody uses... it doesn't need calculus of variations at all. Every step is an elementary calculus step. But the downside is the thought process that actually leads to the derivation is the one you'd only really get after studying a bit of calculus of variations, so you wouldn't come up with it without having that foundation (even though it's not required). (Yes, you can try to remove time from this proof and still keep it rigorous, but then it'd require a stronger background than just freshman calculus in order to be understood.)

[1] The reason I was satisfied with this is that it's obvious that the speed of the particle is irrelevant to the length of the path so long as the medium is uniform, so once I proved it for light of constant speed, I was done. It's not necessarily satisfying for a mathematician though.