I checked the casualty lists before I noticed the date of his death. The Treaty of Paris was signed on 30 March 1856 . . . so he died after the end of the war. That still doesn't indicate how he died, but sickness or disease was the most prevalent cause of deaths during the war. If you are interested in pursuing it any further, his death is probably recorded on the regimental musters kept at the National Archives at Kew. All you need is his regiment and the date of his death and you can check the musters. If you don't live near London (as I certainly don't), some kind soul might offer to check for you. Otherwise, professional researchers should do it for a modest fee.
Based on my experience with regimental musters, how much is recorded seems to depend on the conscientiousness and loquacity of the person keeping the record. Some just record "died" while others indicate the cause.