You have screwed up the situation beyond all repair, in all likelihood.
In all likelihood, you screwed up and rejected a good job offer for no good reason. There is also a possibility that you fell for a phishing scam about which you can't do anything now.
If you are going to ask for advice, ask before you screw things up, and not afterwards.
Perhaps there is a remote chance that you could immediately write back to HR and/or the background check firm and apologize for misunderstanding the situation and salvage your job offer. But, you probably can't.
As I understand it, you are saying that you received an offer letter from a company where you had applied for a job. Then, after you received that offer, you consented to allow the company to hire a third party contractor to do a background check on you.
Then, there are glitches in the processing of the paperwork for which you didn't feel you received a satisfactory explanation. So, you sought to revoke your consent to the background check. Now, you wonder if the means by which you communicated your revocation of consent were adequate.
I will proceed on the assumption that this is what happened and what you are asking. I will also assume that the job to which you were hired was as an employee at will, and that general principles of U.S. law apply as you did not specify where any of this happened and that is the law that I am most familiar with.
You May Have Kissed A Good Job Offer Goodbye For No Good Reason
My first reaction is to point out that if you have withdrawn your consent to a background check that you have just declined to accept an offer of employment that you were otherwise going to receive in all likelihood, because you were upset at how the employer's contractor handled the background check process.
When you receive a job offer and are asked to sign a bunch of paperwork, you aren't free to sign and not sign the paperwork as you see fit. The offer is conditional upon you signing everything without objection. The price of not consenting to any single one of those conditions is that you do not receive the job.
If you don't like their background check contractor, for example, you have two choices. Consent to them anyway, or reject the job entirely. You aren't free to have your cake and eat it too.
Do you realize that you just kissed a job at a very established organization that has not itself done anything obviously wrong goodbye? Withdrawing consent is completely equivalent to rejecting the job offer.
Did you really want to do that?
If you didn't want to reject the job offer, that is really unfortunate, because it is probably too late now to restore the job offer.
Also, nothing in the question indicates that you have any reason to believe that a background check made on you would reveal any negative information. So it is very puzzling why you would turn down a job offer over a reason related to giving permission to have a background check done.
You Could Be The Victim Of A Phishing Scam
The possibility that you rejected a legitimate job offer seems to be far more likely, but there is another much less likely possibility.
It could be that you believe that the glitches involved in the job offer acceptance paperwork are evidence that you find to be credible that the job offer never really existed and was instead a phishing scam designed to get your consent to access your private information.
But, if that is really what happened, why would you think that communicating a withdrawal of consent to have a background check done, no matter how correct you were legally, would prevent a company that already sent you a fake job offer from using your original consent to conduct a background check? A firm that would send you a fake job offer would also have no qualms about ignoring your withdrawal of consent communication, regardless of the means by which notice was given.
Conclusion: It Is Probably Too Late To Fix The Situation Now
So, to conclude, there are really only two possibilities here, either of which make your question about the means by which notice of your withdrawal of consent to do a background check irrelevant.
The first is that you got a legitimate job offer from a legitimate company and rejected that offer out of hand because you were paranoid about glitches in the way documents in your application were processed, even though the background check you gave consent to at the outset wouldn't have done you any harm.
Your best shot is to grovel to HR and the background check firm to say that you were confused and that you are fine with them doing a background check or whatever else they want to do, even though that will probably not work.
The second is that you got a fake job offer and were scammed by a phishing operation which had already accomplished its mission before you contacted them to withdraw your consent, and that criminal operation will now ignore your withdrawal of consent no matter how you communicated it to them. So, if that it true, you need to be vigilant in keeping an eye out for ways that you could be an identity theft victim and may want to change things like passwords and account numbers that the information that could be revealed by a background heck could give them access to.
Either way, this is really unfortunate and you have my sympathy, but there is nothing meaningful that you can do now.