No. Not if you opened the SVG in Illustrator. Once opened, the file is assumed to be live and AI doesn't speak with the file system any further until you save or manually request interaction. Opened artwork is sort of in a "suspended" state - It does not reference the file on disk in terms of "reading" at any point once loaded into AI.
You could try the
Revert command in the File menu... that might work. AI will close the open document and then reopen it from the file on disk. That may cause the loss of any additional information in the AI file if you'd edited there as well.
** -- I make no guarantees that using
Revert won't lose data. It might -- **
If you wish to "refresh" an object in Illustrator it must be a link. You can then refresh the link via the Links Panel. Linked files can not be edited as "live" artwork in Illustrator.
Unfortunately, it appears as though Illustrator simply won't link to an SVG file. When you "place" an SVG into AI, the art is embedded by default. There appears to be no option to "link" to the original SVG file. Sorry.