Difficult to give an actual figure to that question (although some have come up with a figure of 2 billion), as it depends on so many variables. The carrying capacity of each country would need to be calculated, based on available resources and the rate at which those resources would be consumed in order to live to a decent standard without too much adverse environmental impact.
Energy availability, migration, land space for food and housing, water availability, fertility rates, employment levels, etc would all have to be known in order for a sustainable per-country estimate to be calculated.
I don't know whether or not this has been researched on a country-by-country basis.
Globally, a figure of 2 billion has been proposed:
http://populationmatters.org/documents/population_numbers.pdf
"Through the use of a population policy that respects individual rights, and effective resource use policies, as well as science and technology to enhance energy supplies and protect the integrity of the environment, an optimum population of 2 billion people can be achieved."
The report goes on to highlight an important point:
"Historically, decisions to protect the environment have been based on isolated crises and catastrophes. Instead of examining the problem in a holistic, proactive manner, these ad hoc decisions have been designed to protect and/or promote a particular resource or aspect of human wellbeing in the short-term. Our concern, based on past experience, is that these urgent issues relating to human carrying capacity of the world may not be addressed holistically until the situation becomes intolerable or, possibly, irreversible."
Whatever the number for a sustainable population turns out to be, rest assured it will be reached eventually, either because we have decided to act or because Nature has acted on our behalf.
Edit: I've just realised that you were asking about minimum resources required to live, not a sustainable population size. Sorry! Hope someone got something out of my response, though.