I never heard of a "fixed do system" but I suppose it is the system that was used before the "natural" tones a-g were introduced. I don't see any reason to rename the notes a-g back to doremi. Well, French people still use the old system, as "ut re mi fa sol la ti", so that could be called "fixed do system" or rather "fixed ut system".
Actually, most countries other than the United States and Great Britain used fixed-do - "do-re-mi-fa-sol-la-si-do." Part of the reason that it is used is that these are much easier vowels to sing on than A-B-C-D-E-F-G.