There are three premises to this argument and a conclusion. By standard methods there are three parts to an argument:
  1. Premise
  2. Claim
  3. Evidence
Evidence in this case is wholly lacking. Furthermore, the claim must be supported by this evidence. The premises must also be accepted as true.

Furthermore, it is helpful for the argument to show where counter arguments (such as the one that I will make) are flawed. This is often done as evidence because that is one of the two things the counter arguments can attack (the other is the form)

  1. A capitalist economy must expand in order to remain healthy
  2. Expansion requires resources
  3. Resources are finite within the universe

  4. Therefore, capitalism is doomed to eventual failure and collapse is an excellent source for reading on logic and arguments and includes a large number of fallacies.

Hasty Generalization: A generalization based on too little evidence, or on evidence that is biased. Example: All men are testosterone-driven idiots. Or: After being in New York for a week, I can tell you: all New Yorkers are rude.
There is both too little evidence and that which is there is heavily biased.

A capitalist economy must expand in order to remain healthy
First off, just to make sure that everyone is talking about the same thing - let me define capitalism for the duration of this argument. It is unfortunate that this hasn't been done prior to my critique. Capitalism is an economic system which is characterized by private ownership of capital goods, investments by private decision, and prices, production, and distribution of goods are determined by competition in a free market. Thank you Merriam-Webster.

Now that that is defined, the statement that 'a capitalist economy must expand to remain healthy' can be looked at. Unfortunately, I cannot find any papers written about exactly what constitutes a healthy economy - but then I'm not an economist. Still, the question remains "is it possible for a system as described above to be healthy without expanding?" Can a capitalist system maintain a status-quo?

A status quo with respect to what? Here, we look at the second premise, and see that this is referring to resources. Capitalism itself will never hold a status quo for long with respect to ideas - some disruptive idea comes along and throws the economy upside down for awhile until things settle out. This is natural and good in the long run. It is even possible for the long run to reduce the consumption of resources as we see with the emerging ultra-light cars that are either electric or fuel cell powered. These cars take less energy to move around, and do so more efficiently. Why? Because someone can make money selling them. Thats capitalism for you. But yet, there is no evidence that a capitalist economy requires constantly more resources.

Expansion requires resources
There is that resource again. What does it really mean? A source of supply or support, a natural source of wealth or revenue, a source of information or expertise.

First off realize, that everything requires resources of of sort or another. It would be interesting to see what sort of economy exists with the heat death of the universe - if there is anything it will be the hoarding of what little energy can be found.

I've previously attempted to show that expansion of market does not require more physical resources. If anything, in economies where resources are important, there is an increased demand for more efficient use of these resources - higher gas millage, better recycling processes. Look at Japan, a very resource poor nation, and we see extremely efficient use and recycling of these resources. And yet, they continue to expand with better and more effective products.

The resources that are required more of are not the physical type, but rather the human type - ideas. But then, this is not about the resources of the mind. Knowledge is power.

Resources are finite within the universe
There are a finite number of particles in the universe. Yes. Its a really big number. As humans, we haven't even reached Type I civilization on the Kardaschev scale. It is difficult to extrapolate the resources (physical and energy) available to a society that is Type I, much less Type II or Type III (or the hypothetical type IV that can control and use all available energy in the universe), much less the economy that such a civilization would have.

The question of wise use of resources is one of the foresight of the people looking at long term investments. This is not unique to one system over another, but rather the availability of the resources and the forethought of management. I have yet to see research into the infrastructure of energy consumption in communist countries. The wind mill power arrays in the American southwest, the solar power farms. Are there examples of this forethought in Cuba or the old Soviet Union? Still, the lack of forsight cannot tacked on to capitalism - some people have it, some don't.

The incentive for a wise choice comes from looking far enough ahead, not from the economic system a person is in.

Therefore, capitalism is doomed to eventual failure and collapse
Capitalism is no more doomed to eventual failure and collapse than that of any other economic system. All economies and civilizations require some resources There is nothing unique about capitalism in this way. There are possible claims that I an throwing in a red herring, trying to draw the attention away from disproving the above argument - nevertheless...

  1. A (fill in) economy must consume resources to remain healthy
  2. Resources are finite within the universe
  3. Therefore, a (fill in) economy is doomed to eventual failure and collapse.
This is just as good (or bad) an argument as has been initially proposed and would show that any economy will eventually fail. Nothing unique about capitalism has been proven. When there are no resources, there is no economy. It does not matter if this is a capitalistic one or socialistic one.

I hope that I have shown sufficient reason that all of the premises are flawed and that the claim is moot with regards to the distinction that one economic system is doomed while others are not.