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 $3+$.