wget - free open source, command line based, use the recursive (-r) option
Screaming Frog - requires payment after 500 pages, Windows GUI
Of course there are hundreds of other options if you search for "web ...
Have you looked at the Status column? It should show "Success" if Google reads it correctly as well as the last date Google read your sitemap.
Just as an FYI, there's lots of comments here regarding the Sitemap Paradox. It might be beneficial for you to read through that discussion.