Indeed. Fortran has long had good options in matrix exponentiation including the work by Robert Sidje at the University of Queensland with the EXPOKIT library: https://www.maths.uq.edu.au/expokit/paper.pdf
Then there is also the excellent work by John Burkardt: FORTRAN90 Source Codes
And the more recent work by Nicholas Higham (NAG – Nick Higham) on matrix functions including exponentials that Julia also uses and which might be in NAG Library (@themos may have better details) callable in Fortran but perhaps from behind a paywall.
Thus there remain good options that someone packaging up a modern library including with Fortran stdlib might be able to use on the backend.