Living in Shipley and commuting down the LLC to Leeds city centre twice a week i know this section very well.
I would say there are only a few sections which are suitable for a road bike.
Heading out of Leeds you have a fairly smooth track upto Kirkstall marina. In dry conditions this is quite gritty and dusty and would ruin a chain. In wet conditions you'd be slipping all over the place without at least 28s fitted. There are also some rough-ish patches on this section, including some cobbles, stone etc.
From Kirkstall marina to Kirkstall
Morrisons you have a couple of miles of smooth tarmac for which a road bike would be fine.
From Kirkstall Morrisons to Rodley again this is mainly smooth track, but not as easy going as the city centre section particularly around The Abbey pub turn off which is quite rough and can be very wet.
Rodley to Apperley Bridge is completely unsuitable for a road bike. Lots of sections of rough stone that are a pain in the arse to ride on on a any bike (i'm on a cyclo cross bike with 35s).
Apperley Bridge to Shipley is ok when dry but this is not as well draining as the city centre section so tends to be pretty sticky a lot of the time even in summer. The exception to this is the section around Esholt which is quite rough.
Shipley to Saltaire is fine. Not tarmacked but a good smooth surface.
My advise would be to keep a road bike off the canal. It's not what it's made for. Get a hydrid, a hardtail or a cross bike.