Can ITensor be called by Fortran?
In principle yes this should be possible. I believe it’s more or less straightforward to call C (not C++) functions from Fortran. Then it is easy to wrap C++ ITensor code inside of a plain C interface. So you would need to write a small translation layer but nothing that would be too hard. You would not be able to use the whole ITensor interface, though.
An alternative that could be easier is to write the data your Fortran program generates to an HDF5 file, then read this data into an ITensor program to continue the computation there.