For example, if I use some iterative solvers to find a solution to a non-linear least squares problem, is that already considered machine learning?
T. Mitchell defines machine learning in "Machine Learning" book as
a computer program is said to learn from experience 𝐸 concerning some class of tasks 𝑇 and performance measure 𝑃, if its performance at tasks in 𝑇, as measured by 𝑃, improves with experience 𝐸
Hence, based on the above definition, we can't say a machine learning method to every iterative method. In your specific example, it is just a non-linear solver such as the Newton method to finding roots.
However, you should notice that a non-specific machine learning method can be used in the learning process. For example, you might need some numerical methods to compute the measure $P$ (in the above definition). But, we can't say that the specified method is a machine learning method.