My question to you: do you mean legally? Also, it would be appreciated - for the sake of my reply - if you were to narrow down "European" slavery in "the Americans" (which is an enormous topic) and specify perhaps some geographic areas, groups and/or nationalities or time periods.

The simple answer is that the concept of slavery as it existed in Roman times really only has a tangential relationship to what was practiced in the Americas. Roman slavery gave slaves certain rights, and was not based at all on race. Meanwhile, American slavery was based for all intents and purposes entirely on race. The requirements, or perhaps more correctly, the enforced environment, that American slaves generally worked in was abhorrent as compared to that which most ancient slaves would have experienced. Indeed, one of the main reasons for the need for so many slaves was the horrible conditions, particularly in terms of disease, that prevent free European labourers from exploiting the land. The closest comparable I can think of here would be the Roman gold mines in modern day Spain and Portugal that would have had horrible conditions as well. However, there is no evidence that agricultural slaves in the Empire faced anywhere close to the level of brutality and death that African slaves faced in the Americas. The closest American slavery came to that practiced by the Romans might be the 'house slaves', and indeed, the differences here are somewhat appreciable. But I think they are entirely coincidental.

This is a really interesting question. The Roman Republic as an ideal which early Americans were influenced by seems to be common knowledge. If there are no explicit influences it would be equally interesting to know if there are common sociological and psychological influences which the institution of slavery itself had on both Ancient Romans and Americans.