Why do you think free enterprise and capitalism usually develop as economic systems under democracy?

I don't think it is usually that. Certainly the Greeks did not have free enterprise and capitalism when they had democracy. What you are refering to is only recent times.

Socialism is in fact full democracy. A number of Marxists have articulated this.

Your teacher is obviously ignoring Democratic principles. Here, read them for yourself, from the centuries greatest advocate of Democracy: http://teachingamericanhistory.org/library/index.asp?document=2208

The essential difference between democracy and socialism is not the aim, or the core principles, but rather a matter of practical policy implementation....adapted from Woodrow Wilson.

Now a question for you: Isn't it odd how history gets twisted so political views come into and out of vogue? Your question reflects a current popular rightwing fundamental belief, which is erronous to the core.