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For instance, in MATLAB I might use lsqnonlin which implements the Levenberg-Marquardt algorithm.
I found this reference which seems to be useful place to start, but is there also a systematic way to find relevant libraries for Fortran? I know IMSL is a well-known one, but is the general body of Fortran libraries not free as they are for Python or R?
What you found is probably the best list and is unusual in listing the licenses. There are also many specialized programs/libraries out there that can be difficult to find. Many of the libraries were written before people were sensitive to licensing and the licensing terms aren’t as clearly stated as today. Many were government sponsored, making them public domain, though this isn’t clearly stated. If you are making a commercial product the legal issues need to be considered. I frequently check http://people.sc.fsu.edu/~jburkardt/f_src/slatec/slatec.html (on the list you found) because the routines have been brought up to Fortran 90. I also use the GNU Scientific Library via the ISO C Binding. This library is comprehensive and GPL licensed. A Fortran interface library for that purpose is available at http://www.lrz.de/services/software/mathematik/gsl/fortran/.