In the United States there is no requirement to tell the employer who you will be working for, unless there is a specific work contract that would spell out that requirement.
As for the new employer checking on your current employment as part of the background check, that generally not a problem for most employees. Many companies handle all employment verifications via a third party. They will confirm dates of employment only. (For loans they will provide income verification.) You current manager will not be told about the verification check.
In my experience many employers do want to know, as do your co-workers. They either want to know because they care about you, or they are looking for a future place to get a job. But because some people can't be trusted most employees are under no obligation to tell them.
Now HR may ask about a way to contact you. That is so they can send your final check, the tax forms in January or if they find you left something valuable in your office.
Note: don't resign until you have a signed offer letter. If you are still deciding don't resign yet.