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Trecile and Jeff, these replies are not reassuring for the use of a city-bike. What would be useful is to give an opinion based on the quantity and/or quality of the trails. Thanks Gerrit.
So what you're really saying is "I have a piece of string, will it be long enough to tie up a parcel?" - how long is the string? How big is the parcel?
We would need to know your level of fitness, the gear ratios of your hub gears (there
are some steep climbs), the type of tire you intend to be using (my "city bike" runs on almost slicks, my MTB/VTT has tires like a Vibram boot sole), how much gear you intend to load the bike with (are you bikepacking or taking laden panniers?) . . . there are all sorts of variables.
Asphalt main roads in Spain tend to be well maintained, the more rural ones not so much. Concrete surfaces through small villages tend to be "dump it on the surface and roughly rake it around".
Newer rural cycle tracks tend to be well graded gravel in the region of 5 - 8mm (retired highway engineer speaking here), Some are roughly graded with cobbles (4"/ 100mm and upwards) - you can always get off and push. In urban areas you may come across cycle tracks marked out on asphalt roads, through public parks, alongside main rivers.
Unmade up rural roads could be anything - you'll sometimes see pot holes filled with broken brick, tiles, plaster rubble. Sometimes they are just compacted mud or covered in cow dung - during a wet harvest this would be particularly trying, wet clay and straw can clog up your drive train in no time. If your bike has a chain guard and fenders/mudguards you might consider removing them and risking a rooster tail up your back.
You
could do the whole of the
Camino Frances on a Dutch Omafiets, whether you would enjoy it or not is a question only you can answer.
It might be worth your while taking one of the many GPS routes available and overlaying it on Google Earth and following it along at street level and you'll see pretty much what you'll get.
Once again, Buen Camino.