an economic and political system in which a country's trade and industry are controlled by private owners for profit, rather than by the state
Lets refine a bit your Capitalism definition...
Taken by wikipedia:
Capitalism is an economic system based upon private ownership of the means of production and their operation for profit.
State is not related with Capitalism.
And as u pointed out in an abstract way capitalism is perfectly fine with Islam! Actually, Islam is much closer to Capitalism than what Christianity is!
And what u refer that Muslims want is a Caliphate with a Caliph. Historically, the "last" somewhat accepted caliphate was from the Ottoman Empire. After the dissolve of the empire no Muslim nation/power managed to set a caliph that would be widely accepted.
And here we come to the next point, Islam is inherently authoritarian. Islam defines a Supreme Leader setting it as an authoritarian ideology.
ps1: Catholicism also has a Supreme Leader (Pope) and is similarly authoritarian to Islam. Notice that I'm mentioning Catholicism and not Christianity because there are other branches of Christianity that are not so authoritarian.
ps2: To the best of my knowledge only 1 zero authoritarian ideology exists: Anarchism. In general we tend to call authoritarian a system that has 1 supreme leader, or very small leadership that has supreme authority (examples: Stalin's communist party; China's communist party).
Economically speaking:
So Islam, up to the point of "religious restrictions" is completely compatible with capitalism. Actually, more than what Christianity is which promotes the abolish of personal property and promotes its donation to the needy.
Politically speaking:
Islam is authoritarian, and inherently does not go well with Democracy (especially with the Shia Muslims). Sunni Muslims are authoritarian again but from a more wide population base (I would put them close to Catholics in that aspect choose in a representative way your supreme leader).
Do we agree up to here?
Next point, can we have Capitalism and True Democracy together?
Okay, this is difficult to answer. The moment you have Capitalism you automatically create entities that have more power than others. These entities, lets use the term Oligarchs, end up controlling the Democratic system. This results in having ochlocracy (mob ruling) which is what we experience pretty often in modern times.
Okay, I'm taking it far I know... But what I wanted to point out is that in these discussions people are mixing different ideologies (common problem).
Economic Ideologies:
Capitalism
Communism
Socialism
Political Ideologies:
Anarchism
Democracy
Authoritarian Systems (Kingdoms, Dictatorships, Oligarchies, etc)
Religious Ideologies:
Catholicism
Other Christianity
Islam
For many people from the USA: Democracy = Capitalism; Christianity = Catholicism; Socialism = Communism; Communism implies Dictatorship etc...
So you get funny discussions with questions like: Communism is good or bad? But they are actually asking whether a specific communist regime of Russia was good or bad?
Because if they asked specifically whether the Communist Ideology (as theoretically is defined) is good or bad? The answer would be obvious :-P
The problem is whether good ideologies are actually applicable.
Take for example: Anarchism. It is the best ideologically political system... But it is simply Utopian and not applicable. The closest we can get to it is a Democratic Socialist country such as Sweden. Many philosophers/sociologists claim that the next step which should be doable is to go to a Democratic Communist state but we are not there yet.
Ohh and where religion fits to all these? Well you cant have an authoritarian religion to be dominant if you want a truly Democratic Socialist country because you would have to many "authoritarian supporters = rightist". To return to our example Sweden is not Catholic but Lutheran (Protestant) that mean no Pope, no Religious authority (more or less).
I hope my post helps clarify why these discussions lead to nowhere usually :-D