Presidents must undergo presenting their credentials to a host nation when arriving (handled by staff, of course). If they didn’t, could that be considered an act of war?
technically, the president is just a citizen of (say) the USofA entering Canada…
if he has all the required prerequesits (passport, mostly) , he will be allowed in - just like any Joe Doe, otherwise not.
No need to play the WAR-cart so fast 
I agree. If I had the proper documentation and entered a foreign country, it wouldn’t be considered an act of war. So why would it if the individual entering the country under the same circumstances was the head of state?
The premise is that the head of state did NOT enter legally using proper documentation. Additionally, some heads of state don’t have documentation, like Charles III. If the Emperor of Elbonia were to cross the border into Absurdistan without being invited, could it be considered an act of war, if Absurdistan really wanted to make hay about it?
Or did but was denied like Charles III crosses the Canadian border without the permission of the US and calls us “The Colonies”. I know that some anti-immigration people called the crossing of the US border by Mexican foreign nationals (with the tacit approval of Vicente Fox) an invasion by Mexico and felt it was an act of war. That seems a bit extreme IMO.
Historically more likely to be considered an act of abdication/resignation. Or even an act of war against their own people (the implication being they return with a foreign army to squash whatever domestic malcontent they are fleeing from)
No, it’s undocumented entry. The president could be deported but there is nothing about this that is an act of war. An act of war is an attack. Just crossing the border is not an attack.
If the two countries were already at war, and the president of the invading country entered, I suppose he could be shot as an enemy combatant, but that wasn’t your question.
If the Head(s) Of State say it is an Act Of War, then it is an Act Of War. There are no global rules for this situation.
Suppose the President of Ireland takes a holiday in, say, Spain. It’s a private holidy — no official business; he’s visiting family or friends. Ordinarily the Spanish authorities will be notified in advance and he’ll be afforded certain courtesies. But if, for whatever reason, this isn’t done — he’s visiting a lover, say, and wants to keep things very much on the down low — is that an act of war? Of course not. There’s no use of force, no hostile intentions, no violation of Spanish sovereignty. Simply entering a country is, in itself, nothing remotely like an act of war, even if you have some official or representative capacity in your home state.
The US definition of “Act of War” seems to require actual use of arms in some way. However, there are references to an act of war also including a foreign leader to provoke a war. So for the OP the questions are:
For what purpose did the head of state enter the US?
Were they carrying a gun?
Might an unannounced, private visit to another country violate diplomatic protocol?
Probably, yes. The host state will want to know that the VIP is there, if only because they’ll be acutely embarrassed if anything happens to the VIP while in their territory, so they’ll want sufficient advance notice to make appropriate security arrangements. For them not to be told that the VIP was coming would probably seem rude, dismissive, careless. But not a violation of sovereignty and certainly not an act of war.
(These considerations aren’t limited to heads of state; the position would be the same with regard to any high government official making a visit. The amount of notice that it’s reasonable to give, and the appropriate security measures for the host state to take, will obviously vary from case to case.)
Has this ever happened? A head of state or other VIP entering another country without fanfare?
I’m sure that it has happened many times before, without fanfare. Imagine if the Governor-General of Canada (Canada’s acting head of state) wanted to take the kids to Disneyland, and otherwise enjoy a week-long California vacation. It’s hardly a state visit; it’s just a vacation.
However, I would imagine that the G-G, being a Head of State of a foreign country, would get in touch with American authorities, and let them know. That would be the courteous thing to do. “Listen, I’m planning on visiting the US. Not officially, it’s a vacation with the kids.” American authorities would likely say, “Hey thanks for the heads-up. Have fun!”
Without fanfare? Happens all the time. Heads of state visit family or friends or take vacations just like anybody else. Why not?
Without any notice at all? Probably doesn’t happen often.
“Act of war” is not a term in modern international law. The equivalent term would be “aggression” or “armed attack”, and a such an incident might trigger the right of the attacked state to self-defence; in many cases, the Security Council will in its resolutions even avoid these terms for diplomatic reasons and declare a “breach of peace” instead. But either way, the entry of a head of state into another state does not meet this threshold.
Entering a country without permission would not be considered an act of war. Arresting a head of state for entering a country without permission and refusing to release them might be, though.
I’d doubt it. It’d be a violation of international law, for sure, owing to the immunity that heads of state enjoy (unless there’s an exception to such immunity, e.g. if the head of state is accused of crimes against humanity - but the law is in flux here, and the details are disputed). But not every violation of international law amounts to an armed attack that triggers the right to self-defence. The tendency is to interpret the term narrowly and exclude even some incidents that involve the use of force, so as to avoid an escalation of hostilities (in its 1986 judgment in the Nicaragua case, the ICJ referred to this as “a mere frontier incident”).
It’s only an act of war if the leader of the country being entered says:
Canada issues separate passports for people travelling on official business with diplomatic privileges:
People issued these passports may also have a regular passport. The Diplomatic passport is used for official business only, they must use their regular passport for any other travel. The visa requirements may be different between the two.
If they use the regular passport, they are not entitled to any special treatment.