Is it possible he would be listed as "James Davidson"? See results below:
The Medal Rolls for the 93rd Regiment of Foot list:
2779 Corporal James Davidson p. 22 for the Balaclava Clasp 1159 Private James Davidson p. 25 for the same Clasp; Sebastopol Clasp p. 13. 3088 Private James Davidson p. 25 for the same Clasp.
There is also 1855 Private Jonathan Davidson who shows entitlement to a Crimean Medal on page 52. There is a comment by his name which is illegible but looks like it might be a "Regt" with a date, which was frequently put beside names of soldiers who died or were discharged. The only way to check into this further to my knowledge is to have someone check the muster rolls for the regiment during the Crimean War in the National Archives at Kew. If you are nowhere near, I can recommend an inexpensive researcher.