Yes. Assuming the patient is mentally competent to make that decision for himself or herself the patient can request the presence of a family member or significant other during a physical exam. The HIPAA (Health Insurance Portability and Accountability Act) covers a wide range of patient privacy issues, and the federal government has issued numerous guidelines and clarifications regarding the issue you are asking about here. (check out the links below to see the recommendations as they are given to doctors).
For routine matters, the patient's request to have the person in the room is seen as general permission to discuss the patient's health or medical treatments. The one big caveat is whether or not to discuss something of a highly personal nature - HIV, for instance - with a patient in front of the other person, whether they patient invited them or not. In such a case, the doctor is advised to give the patient a heads-up as to what the discussion may entail, and ask whether they want to continue, or ask the other person to step out of the room first.