Believe or not, for the past couple of weeks, Gmail has completely stopped working on my iPhone. I have been scouring the internet to try to find out why this happened. Just now, I finally found a solution to the problem. Unfortunately, according to the docs, it is merely a temporary fix, and the issue will resurface again in the future. However, this fix does work .
For brevity's sake, here are the instructions:
1. On your phone, navigate to the following page: https://www.google.com/accounts/displayunlockcaptcha
2. Now attempt to access you email through the iPhone
Once again, according to the place where I found this information, it appears as though this fix will last for about a couple of months before you have to apply it again.
Pay it forward!!!