The title of this post is pretty self explanatory, but it's also pretty subjective. First, let's define "better" in this context. I don't mean to say that the Natives of America were smarter or more scientifically advanced; quite the opposite. What I'm talking about is the standard of living of the average member of society, as well as the equality of it's citizens (with regards to gender and race). At the point during which the cultural exchange first began between Europe and America (16th century or so), European society was ingrained with an ultra-regressive mentality that included extreme sexism, racism (though in a different manner than 19th century American slavery), classism and extreme religious xenophobia. I'd like to compare and contrast those features with certain Native American cultures to illustrate how horrific the genocide against them was; it's honestly like watching a pulp serial villain ravage an entire continent of cultures.
Before I proceed, I want to be sure I'm not guilty of lumping all of the Natives into the same boat; there were over 150 Native Nations and over 175 Native Languages. Not all of these cultures are going to share traits. On some rare occasions, Natives of the American continent were guilty of atrocities like human sacrifice and slave labor (Aztec society in particular). Because of the diversity in cultures, when I say "Native American", what I mean are Native American cultures from the North and South East areas of modern day United States. Most of these cultures share certain characteristics that the European culture shed a good deal of blood to acquire. The first of these traits is religious tolerance, and it doesn't take very much explanation. Native American religions are not and have never been evangelical; this leads to the conclusion that such religious practice was not forced onto others, as it was in Europe. No Inquisitions, no Crusades, no genocides.
Native American class structure was drastically different as well; rather than a hierarchical system that was based off of feudalism, the class structure was based on group agriculture and tribal hunting and gathering. Although positions of authority existed (Chiefs, etc) they were more positions of arbitration (with regards to which resources are applied where) than positions of ownership; for instance, where a King would own all property on his Kingdom and could tax accordingly, a Chief would own very little (if anything) of his Chiefdom and simply arbitrate the distribution of resource to ensure the community as a whole thrived. Where a King would trade with neighboring kingdoms for personal profit, and tax trading between his denizens and neighboring nations through tariffs, the Chief would initiate and negotiate trade for the benefit of the entire community.
Racism was nearly non-existent in Native American society. Indeed, when Jamestown colonists began dying by the truck load during the Starving Times, many defected to live with the Natives rather than continue to live a harrowing existence for the profit of the Virginia Company. "Race" was not established as a societal construct in Native American culture, whereas within European society, it was already strongly established and used as a basis for cultural assessment. I could elaborate quite a bit about the racism in 16th century Europe, but I don't think anyone needs an explanation of that racism or why the Native American's alternative is superior.
Perhaps the most poignant difference between the two societies is their treatment of gender roles. This includes both male/female relations and homo/heterosexual interactions. I'm not going to imply or suggest that the Native Americans had no sense of gender roles; men were expected to hunt and women expected to gather. This wasn't set in stone, however, and anyone breaking gender norms was labeled a niizh manidoowag, or "Two Spirit" roughly translated (Europeans called such Natives "berdaches", a highly offensive term). It's clear that the Natives had a much more embrasive view of these members of society and often had four or five "gender roles", rather than the European roles which include Male and Female (with anyone not conforming to these norms being executed). It's clear that this threatened the European culture and was one of the first things that America targetted in it's attempt to assimilate or destroy all Native American culture (such as the Dawes Act which required Natives to declare a "head of household" and assemble in a European nuclear family, with a male as the sole property owner and the superior to his wife).
Put in the most general terms, Native Americans were much more flexible with thier social norms than the Europeans. Though they were systematically eradicated (by European use of superior scientific knowledge in the fields of warfare and naval navigation), Natives were much better suited for the average member of society to avoid exploitation and have a high standard of living. Modern day racism, classism, sexism and homophobia would be non-existant if not for the genocide committed against the Natives of this continent.