I don't generally go around thinking about whether someone else is English, I just go with what they say.
I'm a mixed race immigrant. I'm well aware that there are people who will never see me as British because I'm a naturalised citizen and no British blood unless we go back a few hundred years with some questionable documents, too far back to likely remain in my genetics. There were also many people who discussed with me how they thought I was British well before I was legally, based on how long I'd lived in the UK (since I was a teenager) and how I live. I don't mind as long as people aren't asses about it. Someone else not considering me British doesn't bother me.
I've four kids born and raised in England, my husband/their father is English. One identifies as English ethnically, and even then, it's only on paperwork and - in his words - 'because that's how people tend to see me' as he's on the lighter side of medium skin tone. They've all met people who've told them they don't consider them English or British because of me or how they talk or because we don't do some specific English thing. That's part of why three of my kids don't identify as English.
Different groups and individuals have different requirements and those can change over time. I mean, many people are saying people are British if they're born here, but the UK hasn't had jus soli since 1983. Fathers weren't able to pass on citizenship without being married to the mother before 2006. That' citizenship that has laws and usually paperwork. Heritage is more complicated than that. Many groups requiring maternal and/or paternal direct descent to be 'official', and many people - me included - have gaps in what we really know about our heritage.
But to answer your question, they became fully American after the 1924 Indian Citizenship Act, because ‘American’ is considered a nationality, not an ethnicity.
Depends on what you mean by 'fully American'. Many would argue that 1978 with the Indian Child Welfare Act meaning parents could no longer be forced to send their children to residential schools (though still have a higher rate of having children taken into care, and until the 90s this meant being sent to the same schools we've recently found mass graves) and the American Indian Religious Freedom Act - prior to that, it was legal to ban sacred practices including dances and religious attire, though again, there is still a fight on these to this day.
Some Americans view American as both an ethnicity and a nationality. It is included in ethnicity/ancestry option on some US Census surveys and other forms. Much like in many other countries, there is some status in having a long American heritage. While varying by region just like any other country, plenty of Americans have shared heritage, collective history, and cultural traditions, even when it's a genetic and social mix - again, much like most other nations when we go back over the centuries.