If it's a raising bridge, then it must have floor beneath the entire edge along which it raises. Only retracting bridges can be built entirely over open space.
Is it possible to build a raising bridge based upon a set of walls (i.e. no "official" floor underneath the raising side, just "tops of wall"-type foors) then remove the floor without the bridge collapsing?
Yes.
On the edge of a megaproject digging area, I set up a line of walls, and on the level above some access floors to allow me 'building access' to place a raising bridge, based atop those walls. Bridge built, and linked to a lever, I then removed the access floors (not necessary, but did so anyway) and could then remove the walls from below (leaving one at each end of the 'bridge base', but I suspect I only needed to leave one of them, and maybe have gotten away with none at all if there was still an adjacent supporting wall to the side to 'hang off of') leaving a fully functional lever-operated bridge that would nicely act as a "non-stick" wall-type barrier to help martial liquids along a similarly 'non-stick' bridge floor, to meet at a point where obsidian could be spontaneously created without any supporting landscape or structure. The bridge raises and lowers without losing integrity (a further fear that I had, in that it would suddenly 'realise' that it was no longer supported).
The 'wall' bridges would be anchored either side of the 'floor' ones, pointing outwards (or being just one-tile thick and and raising either inwards
or outwards, given that still gives a "wall" when raised). The 'floor' ones (not needing differentiation by raising direction or retraction or lever linking, seeing as they are used "flat", unless also being used as "valves" for the magma or water) would be supported at opposite walls, going between the two wall-like barriers to leave the 'formation gap', thus:
###########
## ##
.. .. Shaft dug with two input routes
## ##
###########
###########
## ##
..|-| |-|.. Two short bridges built to nearly bridge the gap
## ##
###########
###########
##+++++++## On the level below, build walls to create
..|-| |-|..
##+++++++## floors sufficient to build upon.
###########
###########
##|-----|## Upon these wall-tops, with build access from the existing
..|-| |-|.. bridges, build new bridges (set to raising towards/away from
##|-----|## the centre, along their 1-tile thickness).
###########
###########
##|-XXX-|## Deconstruct the now covered-over lower walls
..|-| |-|.. from under at least the middle three parts
##|-XXX-|## of the bridge length, as indicated by "X"s.
########### (And of course any other 'infrastructure'.)
###########
##|=====|## Raise those bridges
..|-| |-|.. to form confining walls.
##|=====|## (Probably not order-critical with previous step.)
###########
I had no magma/water at hand to easily test the operational aspects, but the construction method definitely appears to work. I can't think how one would 'hold' a piece of obsidian in place, so it's not so much "reloading" as "produce-and-drop on demand. Give or take the response time and flow-rate of the supply fluids.