I'm currently working on the code on my Github account, at pemrich / Simple-Google-Calendar-Widget
After digging into the code and doing some searches on the WordPress Forums and the PHP documentation, I have a very good idea of what's wrong and a very loose plan on how to fix it.
After all is said and done, I'll be updating my copy of the code to use Google Calendar API v3 (instead of v2, which it uses now). Bloggers will need a Google API key and will need to enable Google Calendar API. I'll update the instructions both in the Widget and in the documentation for those who don't know how to go about getting their API key and enabling the Calendar API.
Not only will this help me resolve the issue, but it will provide me with additional control for options I want to incorporate later.