Good heavens, there's a fair amount of misinformation on this thread!
In the wild, it is true the majority of fish families are more or less predatory, eating either other fish or various invertebrates, typically insects in the case of freshwater fish. However, several families of fish are more or less exclusively herbivorous. This isn't to say they won't eat meat if offered it, but that in the wild it forms a minor part of their diet and doesn't supply most of their calorific intake. Marine examples include damselfish (which farm algae), sea chub (which eat brown algae), and parrot fish (which eat coraline algae). Freshwater herbivores include tilapia (general plant debris), pacus (fruits and seeds), mollies (green algae), cyprinodont killifish (algae), and some loricariid catfish (algae). The desert pupfish is perhaps the best-known "extreme" veggie, in that it lives in water so hot pretty much nothing else lives there except brown and green algae, which these fish eat. In fact, without algae, it is difficult to maintain in aquaria.
Few of these never eat animals. But then, cows and sheep eat small animals while foraging, and that doesn't stop them being herbivores, while pigs will even catch rats and mice. But tilapia are routinely raised on plant protein on fish farms (this is done to avoid problems with mercury from fish meal) and Panaque spp. are probably harmed by given animal foods and are best kept on a strictly vegetarian diet of wood and vegetables. Panaque have symbiotic bacteria that -- uniquely among fish -- allow them to digest wood.
As a general point, it is probably unwise to try to impose choices we can make (as omnivores) onto aquarium fish that are predominantly carnivores. But if you selected the right species, and made the committment to create the lots of algae in the tank and provided alternative protein sources such as frozen peas, there's probably no reason why you couldn't keep algae-specialists like mollies, Panaque, twig catfish, Florida flagfish, and so on with success. The tricky thing is that you would need to produce a lot of algae to produce the same amount of protein as, say, a portion of daphnia or bloodworms. This would demand strong lighting and periodic fertilisation of the water with minerals. After all, the fish will only be as healthy as the algae they eat. You'd also have to accept that the tiny animals in the algae (rotifers, nematodes, tardigrades, and the like) would be part of their "meals" as well.
Basically, this isn't something I'd recommend, but it would be possible.
Cheers,
Neale