First, income inequality is not a "yes" or "no" condition -every society has it, save for a theoretical absolute communism which no country has ever been close to implement. So America has it now, had it in the past and will have it in the future, same as Ivory Coast, Cuba and Sweden. Second, strictly speaking, an increase in income inequality does not necessarily equal an increase in poverty -we can conceive a scenario in which the total income increases enough so that even if its distribution is more unequal, the number of poor people decreases.
That said, in the vast majority of cases, an increase in income inequality DOES increase poverty. While total income may increase to some extent, the majority of it goes to the hands of those who sit at the top, so the poor remain as they were at best. And since absolute increases of total income are hard to quantify -due to inflation, exchange rates, different standard of living, etc.- in most cases an increase in income inequality is a mere redistribution of resources, which effective increase the number of the poor.
A fairly clear case is that of Brazil: during the 1950s and 60s, the economy of Brazil grew at a really strong pace, yet as its population grew most of them went to increase the ranks of the poor, since they were pretty much excluded from wealth distribution. Social democrat administrations have taken much heat for slowing down the overall growth of the economy, but have also slightly improved the distribution of wealth, effectively reducing the number of poor Brazilians (although there is a lot to be improved there)
No, income inequality does not increase poverty in America, because everyone in America is rich compared to those in other countries. Income inequality results from people who spend money irresponsibly, or from people who become too dependent on government assistance, rather than working. When wealthy people have extra money, they will spend it, and that will help people of all incomes.