The 18th century saw a rise in the use of mathematical formalisms to account for natural phenomena. Works of Lagrange, Euler, d'Alembert, etc., were groundbreaking in the history of mechanics and mathematics. I know there is an expansive literature on mathematical explanation today; but this 'mathematicization' in the 18th century must have also been met with supporters and dissenters. Was there a 'debate' about the role of mathematics in physics in the 18th century? If so, who were the key figures, and what did they claim?
This quite difficult question seems to place the mathematization of nature in the 18th century, but a substantial body of scholarly opinion writes of it as developing earlier than that, in the 17th -- although there can of course be no doubt that its further development in the 18th century was very extensive indeed.
Sources on the 17th century earlier developments in this mathematization include
As for the 18th-century developments, they can be seen as quite heterogeneous in nature and it is possible that a satisfactory history of them has yet to be written.
Newton's work played as might be expected a considerable part in the 18th-century developments and controversies, but it can often be rather difficult to characterize some of the participants in the debates as either Newtonians or opposed, since they would typically accept certain aspects of Newton's work while adopting approaches otherwise opposed to his. There are more than traces of Cartesian abstract thinking especially in subjects hard to reduce to experiments and reliable mathematics: on a Cartesian pattern of thought, acceptability of ideas could depend at least in part on how far they were 'clearly conceivable' or on the other hand 'obscure'. This is arguably an opposite to Newton's severe requirement to "argue from Phænomena" and to subject everything to experiment or observation even where the results or implications were indeed difficult to conceive.
An example may be seen in d'Alembert: although he carried out substantial investigations that relied on Newton as starting-point, he also wrote a Traité de Dynamique (1743) that tried to account for dynamics without the conceptual difficulty of a concept of force. There is a not dissimilar tension in some of Euler's early work in mathematical physics.