The inability of Africa to unify is a function of poor politics, of states with repressive governments. The success of the United States was that the colonies willingly separated the powers of government between the state and federal level through the Constitution (of course, there are those who would argue the federal aspect has grown unchecked at the state's and people's expense, but that's another issue.)
No such willingness to have any dialogue regarding the relations of states exists among the African nations, even if it were to be to their mutual benefit. In order for Africa, with its huge resources, to economically thrive, the governments need to change (or be changed.)
Are you asking if Africa can become one united country? I don't think this is likely. The countries are too busy fighting within themselves and have often been fighting each other. However, if they did decide to unite it would be more like the European Union I imagine, a loose coalition and not one country.
Are you asking about this on a political level or on an economic and social level? I agree with the previous posts about politics. As far as the economy goes, if African countries ever get to the same level as the US, it will take a very long time. The continent as a whole lacks things like infrastructure and stable governments that are needed for true economic development.
When America was formed as the United States, there were not well established individual countries on the continent. America was a group of loosely held colonies. These colonies came together and formed a central government. Eventually, the country came to include 50 states and several territories.
Africa is a very different place. The African continent has many well established countries. Each country has its own government and body of laws. It would be very difficult to encompass these disparate countries under one central government.
I agree with e-martin that Africa could adopt a confederation or some other limited unification. However, I do not see this as a likely scenario.
America's central federal government functions as a potent unifying force in a way that seems unlikely in Africa given the long history of both nationalism and tribalism on the continent.
I don't doubt that Africa could adopt a confederation and unified trade policy however, like a looser version of the EU or perhaps a bit like NATO.