And they'd probably be assimilated into the American Indians and become a part of whichever group in question's lore and mythology. Since the Romans knew about the Canary Islands but had little interaction with them, we can assume they'd have even less interaction with the New World. There is no reason to try and colonise a land so distant when better lands (Scotland, Ireland, Morocco, etc.) are nearby. We know from OTL that the American Indians would probably be like the Germans in resisting Roman influence.
But I guess the Europeans of Antiquity would have plenty to interest American Indians, so a trading colony is possible. It's just with sailing technology at the time, they'd need better ships to be able to regularly supply it with goods. American Indians will want metal tools for farming and daily life, metal weapons, alcohol, etc., as they did OTL. The success of the Greeks (the most likely to make this sort of colony) or Romans depends on their ability to supply this colony. If contact is ever lost, I think they'd be absorbed into the local populations.
That quite possibly happened OTL, albeit probably moreso with Brazil instead of the modern US going by the ocean currents. We know for a fact it happened multiple times with Japanese/Chinese ships on the West Coast. If we dug up enough East Coast/West Coast Indian graves (smallpox victims and such) and gene tested them all, we'd no doubt find Asian or European genes respectively where they shouldn't be found, like how Polynesian ancestry is known to exist in a certain Amazonian ethnic group. They have no hope of preserving their identity, and they will be absorbed into the local peoples. The odds are extremely high they will never see their homeland again, and even if they did, why would the homeland ever send more people to this place instead of treating it as a interesting story and leaving it at that?