I recently failed a drug test with my employer. After the company and myself were notified I was terminated. While talking with the company president of human resources I asked that this matter be kept confidential. The next day I received many calls and texts from former workmates. It was quite apperant the had all the details of my termination. I spoke to nobody about this matter. Does the company have the right to disclose this information with to others without my consent?

You have no right to this kind of privacy unless it is specifically written into company policy.
A company has every right to tell other employees (and future references) why you were fired, especially when that reason was a direct result of your violation of the terms of your employment.

If any information was wrong, that would be a different situation (might be difficult to prove).
If they talked about private matters of your life outside of work or in the past such as when a background check was done, that would be different.

Assuming you're in the US, generally you have a right to privacy where there is a reasonable expectation on it. There shouldn't be hidden cameras in locker rooms or bathrooms because you have a reasonable expectation to privacy there. Health information is protected, even in the workplace, by HIPAA. The employer shouldn't be telling people about the results of a drug screen, but I really doubt you have a case (especially if it's true).

osiris1, very little health information is protected in the workplace by HIPAA.
That isn't even the question here.
To say 'The employer shouldn't be telling people about the results of a drug screen, but I really doubt you have a case (especially if it's true)' is not helpful or meaningful. If an employer 'shouldn't' be doing something then why doesn't this person have a case? There is no law that protects him, and if you can find one, please publish it here.