You can try other apertures, such as f/2, f/2.2, and f/2.5 to see if they give you the Depth of Field (DoF) you want. If you want to keep the background blurred after stopping down then increase the separation between your subject and the background.
If that doesn't solve the problem, you may be experiencing a front-focus issue. If the parts of the face behind the eyes were in better focus than the eyes themselves, it might be a back-focus issue. Unfortunately, the 600D is not capable of user controlled Auto Focus Micro Adjustment that is available in higher end bodies such as the 7D, 70D, and 5D3. Your camera and lens together will need to be calibrated by a Canon Service Center if they need to be adjusted.