A store can insist that a person leave. Why? A retail store is private property. If a person refuses to leave when asked (the classic definition of causing a scene), it is expected that the police will be called and an arrest can be made. A person can be asked to sign a document, legally. If the person refuses, then consequences follow.
It sounds like, contrary to what you posted, that your son was "bothering" someone because he refused to leave when asked. And, by doing so, security and the police had to become involved, which also is "bothering" someone.
Your son was on private property so they can ban him for being there. Different rules apply from public to private property.
As far as forcing someone to sign something being a violation of rights. If you are talking about a violation of Constitutional rights, NO. Typically, Constitutional Rights apply to government entities, here, the store is not a government entity thus no violation of rights.
The prior two answers are reasonably accurate given the information you have provided but I think you may require further clarification. First, you mention that he was arrested but I am not aware of any criminal charge called "obedience to an officer." Presumably, if he was released to you as a minor and not taken immediately into custody.
Ordinarily, the juvenile district attorney would be able to tell you if the store owner filled out a juvenile complaint form and whether or not the DA was going to endorce a petition and schedule a court date. If so, the DA would provide you with an Incident Report written by the arresting officer as part of the Petition and that Petition would state exactly what offense your teenager was being accused in detail.
Now, I believe what you are referring to is what is called a "trespass warning". Here's an excerpt from Georgia case Love v. State (2010), although there are numerous case history on the subject of "trespass warnings" in general.
Georgia code section 16-7-21 (b), provides, in pertinent part, that “[a] person commits the offense of criminal trespass when he or she knowingly and without authority: ... (2) [e]nters upon the land or premises of another person ... after receiving, prior to such entry, notice from ... an authorized representative of the owner or rightful occupant that such entry is forbidden." Our Supreme Court has held that, "[n]otice is an essential element of the offense of criminal trespass, and must be proven by the state beyond a reasonable doubt at trial. Inherent in the statute's notice provision is a requirement that notice be reasonable under the circumstances, as well as sufficiently explicit to apprise the trespasser what property he is forbidden to enter."
Notice is a tricky legal issue sometimes and most case law revolves around whether or not the defendant received adequate "legal" notice that future entry in or upon private premises (here a store) would result in prosecution for criminal trespass. Usually, at trial or motion to supress, that proof is in the form of testimony by security guard or law enforcement officer that the notice had been given.
Here it appears that your teen was asked to sign what appears to me to be a "trespass warning." It was his right to refuse to sign it, but the consequense was for the officer to arrrest him. My guess is that the officer and the store owner would testify that they had given your teen prior warnings. My guess is that the charge will be criminal trespass, not the "obedience to an officer" language you mention.
My advice to you is to tell your teen in no uncertain terms that the store and its environs are permanently off limits that you may not have the money to pay his bond in the future, or his probation fees should he be convicted or plead out.
If your teen gets belligerent with the office at the store in the future he may end up with an obstruction charge as well. This is a no win situation.
I believe previous answers suggested that you find out what's going on with your teen and attempt to resolve it, before your teen ends up with a criminal blemish on his record that may effect his employment opportunities in what bodes to be a very difficult future.
Best of luck!