Can a food processor liquify vegetables and fruit?

If you want a clear, pulp-free juice, you can take the mash from the food processor or blender (which I think would work better), put it in a dish towel/multi-layer cheese cloth/jelly bag held over a bowl, then squeeze, twish, squeeze, etc, to separate the pulp from the juice. Pulp can be used or composted, depending on how edible it is. Even a modest juicer will do the job in no time, though, and you can do drinkable portions easily.

If you want "liquid", then consider using a juicer. Personally, I use a blender in small batches and put through a strainer. Extra work but the final result is superb.

I find that if you process in small batches you're more able to get rid of the chunks. Straining may be unavoidable.