There are multiple problems with Gates's predition, but two of the biggest are these:
1. Growth in real economic wealth is very strongly tied to growth in real resource consumption. The master resource is energy, but numerous other resources are in tight supply, with a critical set being "bauxite whose production peaked in 1943), copper (1998), iron ore (1951), magnesium (1966), phosphate rock (1980), potash (1967), rare earth metals (1984), tin (1945), titanium (1964), and zinc (1969)" (from Richard Heinberg's The End of Growth[1])

I've explored the concept of decoupling in greater length using Wolfram+Alpha data to show the relationship between energy use and GDP for the G8 nations plus China, India, and Brazil, as well as global growth, in the periods of 2000 - 2010, 1990 - 2012, and 1980-2012 (not all data available for all periods, though the 2000 - 2010 data are complete for all nations analyzed). While there's some sign of very weak decoupling of energy and GDP growth, principally in Japan and the USA, for global GDP growth, there's a very strong relationship between GDP and energy usage, and both have been increasing. With limited exceptions, global per capita energy use has also been increasing.

http://www.reddit.com/r/dredmorbius/comments/1vlksg/economic...

As I write this, I'm listening to a news story that the IEA has announced that US oil consumption, flat for years, is up 2%

http://www.marketplace.org/topics/world/rising-demand-oil-gr...

2. The second major problem is that the so-called Demographic Transition which Gates and Hans Rosling like to trumpet is little more than a largely unexplained phenomenon observed in some but not all data series. Tom "Do the Math" Murphy, UCSD physics professor, has specifically looked at this with regards to oil states, and makes the observation that "surplus energy makes babies"[2]. This is significant for two reasons: it means that the demographic transition isn't being observed in all countries, and it means that population growth, and hence domestic energy consumption growth, is highest in the major oil exporting nations. Growing domestic consumption means reduced availability of energy for export markets -- a phenomenon known as the "export lands model". Other research suggests that the causality link between development rates and birth rates is less clear than popularly portrayed[3].

I could bore (or terrify) you with numerous other challenges: flat or falling agricultural productivity, EROEI deficiencies in virtually every non-fossil energy alternative, pandemics risks. There's a reason I don't get invited to parties much .... But I think these two will do.
While I have respect for some of Bill Gates's work (and I'm by no means an uncritical fan of his), his optimism here seems misplaced and founded on a very incomplete portrayal of the situation.

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Notes:

1. Sources: http://minerals.usgs.gov/ds/2005/140/ and http://www.culturechange.org/cms/content/view/597/2/

2. "The Real Population Problem" http://physics.ucsd.edu/do-the-math/2013/09/the-real-populat...

3. "Revisiting demographic transition: correlation and causation in the rate of development and fertility decline." http://www.iussp.org/sites/default/files/event_call_for_pape...