**NO**

Consider points forming a symmetric parabola in the $x$-$y$ plane. Many machine learning models, including a linear regression using a quadratic term, will catch this relationship. However, the Pearson correlation will be (correctly) reported as zero.

A reference could be example 4.5.9 in the second edition of *Statistical Inference* by George Casella and Roger L. Berger, page 174.

Further, once you have multiple variables like you probably do in a machine learning problem, it is not clear what a Pearson correlation between all of them would be, since classical Pearson correlation inputs two variables, not $3+$.