French uses "the" at times when English does not and you cannot translate word for word. http://french.about.com/library/weekly/aa082401.htm
No, he has an illness that affects his heart.
lovesick translates as "chagrin d'amour" heartsick translates as "ayant la mort dans l'âme"
I used the related "he has a heart problem" and got it wrong. It sounds more natural to me, though the French word "probleme" does not appear. We would not say "sick heart" to represent a real heart problem. And the French word for weak (faible) does not appear either. So I would conclude this should not be translated word for word, and my translation is as good as any others that are here, so I'm reporting it.
' He is heartsick ' or ' He is sick at heart ' refer to the spirit, not the body. Googling a sentence is probably a good way to check whether it's common usage. "He has a bad heart " got 45,000 hits; "He has a ill heart " only got 3, one complaining about it in Duolingo and two in what looked like Indian melodramas.
"He is sad." would be "il est triste." http://dictionnaire.reverso.net/anglais-francais/He%20is%20sad