Not necessarily. Between the Psalm 83 and Ezekiel 38 and 39 wars, Islam will be absolutely devastated as a credible, viable "religion".
Last Sunday I spotted J. R. Church's Hidden Prophecies in the Psalms in the church library. Being interested in the history of the current dispensational interpretation, I consulted him on Psalm 83. Church, at least in this edition, equated that conflict with both Ezekiel 38-9 and Armageddon.
(Others reading this might profit from reading Har Magedon: The End of the Millenium.)
The link you gave goes to the Amazon site where the book can be purchased and gives no details of what the book actually says.
The Ps 83 War will allow Israel to destroy Damascus, get rid of Hamas and Hezbollah, and conquer the surrounding land (Lebanon, Jordan, parts of Iraq and Saudi Arabia...). Then, and only then, Israel will live at peace, setting the stage for the Ezekiel 38 war. Both wars take place before the Tribulation, Armaggeddon takes place afterwards. So, how could anybody equate all 3 conflicts?