There are two separate issues 1/ his frequent calls 2/ him feeling sad/depressed.
His possessiveness may in fact relate to his mood. The phone calls may be his way of distracting himself from how he feels. This behavior may also relate to his self esteem as well as possessiveness issues. The behavior may have become addictive (if he cannot restrain himself from calling her over and over)
When you speak with him, you can point out to him that his girlfriend had expressed her love to him and that the numerous phone calls can be tiring to deal with during the day. You tell him that if he was receiving calls like that he may become stressed out as well. It also implies that he does not trust her and that can be a turn off to her. At the moment she may not be doing anything about it but over time, it can become overbearing and she may react to him negatively as a result of it.
You would also want to touch on his mood during your conversation. Ask him if he's noticed a connection b/w his mood and the time of calls he makes during a given day. Speak to him about the fact that if he is set on receiving replies from her to all of his calls, the anticipation of waiting for her to respond can increase his anxiety level and affect his mood. He would then find himself stuck in this cycle, instead of feel better. The two of you can talk about what he would like to do to address his depression and then, you can encourage him to consult with a professional because depression is a treatable condition. Since he's coming to you for advice, it seems that he trusts your judgment and you can remind him that your feedback is objective and intended to help not criticize him. You'd also remind him that he'd want to consider what you're telling him not just ask you and do nothing after that, or do the same thing again. Another thing to point out to him may be that a woman wants to feel special. The more calls he makes out to her, the more she'd become used to him doing this and it would take the alway from the excitement of the chase when his behavior is so predictable.