Perhaps the best example of the US being a melting pot in the time about which you are asking can be found in the very existence of the term "melting pot." This term came to be widely used in the US after the 1908 debut of a play by that name by Israel Zangwill. The fact that a person who was himself a Jew and an advocate for immigrants (though he was born and lived in England) shows that the US was seen as a melting pot country by the sorts of people who came here.
Another example is the fact that immigrants were often able to get started in American (typically city) politics. Many immigrants became involved in labor unions or in political associations that were meant to advance immigrant interests. This shows that the immigrants were becoming American in the sense that they were learning to be part of a democratic society.