it is not, exactly.
'Constitutional Carry' is the nickname given to a policy that used to be known as 'Vermont Carry' in that a law abiding handgun owner had the law on their side if they chose to carry either openly or concealed without a license.
It was known as Vermont Carry for many decades, due to Vermont being the only state with this policy until recently.
One of the first states to follow this route in recent times was Alaska, which until then mandated open carry only and had no provision for concealed. That could be messy with their weather, many folks simply wanted to 'cover up' the handgun due to protecting it from the weather. They can do that now without a license.
Vermont and Alaska treat Constitutional Carry differently in that AK does issue licenses for the purpose of reciprocity (travel to other states which recognize out of state licenses) as opposed to VT which does not issue licenses.
In my region (northeast), Maine has gone Const. Carry recently, yet New Hampshire has declined to do so. NH does have a non-resident carry license available by mail, however.
The vast majority of states in the US do now fit the description in the OP - this is known as 'Unlicensed Open Carry' (UOC) - still most of the states with UOC do require a license for concealed.
There are very few states that do not have some provision for open carry - as a matter of fact it is only about a half dozen or so.
Florida - which was one of the first states to begin the current trend towards a more carry-friendly atmosphere, still prohibits OC, even with a license, I cannot imagine mandatory concealment in that heat.
New Jersey and Maryland - are so anti-gun that anyone lucky enough to get a license should only carry deeply concealed in order to avoid alerting anti-gun civilians or even police officers for that matter.
California - is a patchwork of rules depending on the friendliness of the local police bureaucracy.
Now York - open carry is ill advised unless one is deep in the woods during hunting season when guns are all over the place anyway.
As for the question - I always advise anyone who carries for self defense to seek out training to approximate the level of a law enforcement officer (50 to 100 hours of training) and to qualify on-record each year. This can be done in many states by fulfilling the requirement for armed security through a reputable training organization or the local LE agency.
EDITED to add: DO NOT bring a handgun into NY, NJ or MD without a license to carry in that state. There is NO reciprocity and a felony charge will result. Disclaimer: I Am Not A Lawyer
= new reply since forum marked as read