The 2nd Amendment says that the right to keep and bear arms "shall not be infringed".
Infringement means "A breaking into; a trespass or encroachment upon; a violation of a law, regulation, contract, or right."
Keep means "a term meaning to hold, to maintain, to support, to retain in possession and to take care of."
Bear means "To support, sustain, or carry; to give rise to", or to produce, something else as an incident or auxiliary."
Shall means "As used in statutes and similar instruments, this word is generally imperative or mandatory; but it may be construed as merely permissive or directory, (as equivalent to "may,") to carry out the legislative intention and In cases where no right or benefit to any one depends on its being taken in the imperative sense, and where no public or private right is impaired by its interpretation in the other sense."
So long as the government does not break into or encroach upon your right to keep or bear arms, any action they take would be constitutional by legal definition. No right is absolute. All rights must be balanced against the rights of others.
For example, you have the right to freedom of speech, but you do not have the right to trespass on my property. Therefore you have no freedom of speech on my property. This comes up quite frequently in internet discussion groups and blogs, where people constantly complain about being censored on private fora.
Similarly, you would no right to keep or bear arms on my private property. This is why the legislature cannot mandate that businesses and private universities allow carry in their establishments. However, they can, and have, required that businesses post their desire to prevent your carry in a specific manner.
You have freedom of speech, but you do not have the right to endanger other people with your speech. Thus the famous dicta that you cannot yell fire in a crowded theatre.
Similarly, you have the right to keep and bear arms, but you may not do so in a manner that is threatening to or endangers other people. The question is obviously raised, who decides what is threatening to or endangers other people? The answer is the people do, through their representatives. So, if you want "constitutional carry" (or more accurately unlicensed carry), you must convince the people that it will not endanger them or threaten them in some way.
The Constitution preserves the advantage of being armed which Americans possess over the people of almost every other nation where the governments are afraid to trust the people with arms. James Madison
NRA Life Member Texas Firearms Coalition