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Is it possible to derive the EOM of an inextensible cantilever beam without using any kind of variational principle I mean is it possible to derive it from Newton's law only?

Note: of course This should be the nonlinear case i.e large deflection case

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    $\begingroup$ this is a solved problem. google on free vibratory response of a cantilever beam. $\endgroup$ Commented Jul 1, 2021 at 5:51

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Yes. You can take a differential element and do a balance of forces and moments. Then, apply Euler-Bernoulli hypotheses equate that to mass times acceleration and get what you want. Some of that is done in the Wikipedia article.

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