Firstly, it sounds unnatural to refer to "it" before introducing the word "cold" in your sentence. Secondly, the word "by" is unnecessary. I would phrase the sentence as such:
He got a cold and wanted to cure it himself.
You might also use the word "insist" to emphasize how your subject didn't want to take other people's advice.
He got a cold and insisted on curing it himself.
To make it clear that the subject wanted to get better without the use of medicine, you might also say:
He got a cold and insisted on letting it heal naturally.
Another suggestion from snailboat's comment, which makes it absolutely clear that there was no treatment is involved:
He got a cold and insisted on letting it run its course.