This brings up the Contact he is associated with and I know you can get to Contact Roles there, but my client wants to be able to find the Opportunities directly and go into them without any extra clicks. Is this possible?
No this can't work with the global search out of the box. You can't search for elements in related lists and bring up their parent/master as a match.
You could create a custom text field on opportunity and create a scheduled job (unfortunately triggers are not supported), which concatenates the contact names into that field. You need the carefully decide between text (255) and long text (128000). Both have pros and cons.