Hey there! As always, this is a disclaimer that there are people here who know way more about English history than me-- I am Shakespeare scholar, so I look at Shakespeare through a dramaturgical and theatrical standpoint, and use history for context. With that being said, let's go!

Short answer, yes!

Longer answer: Jokes about the Welsh had to start somewhere, but let's do some textual contextualizing because this is AskHistorians and I'm a nerd who has had at least 2 glasses of wine! To be honest, this is a hugely broad question that deals with all sorts of English culture, from specific national identity to the origins of language to literally questioning the "feeling" and meaning of being British, so I'm going to answer it as best I can. This will have some very, very broad generalizations, but I will do my best to paint an accurate picture towards an answer to your question.

The "real" Henry IV (Henry Bolingbroke) lived from 1367-1413. Even by this point in history, the English considered themselves superior to many of the other Anglo Saxon tribes, particularly the Welsh and Scottish. In fact, they worked...pretty hard to be pretty big dicks to their closest neighbors, which they excused and promoted as a chance to help the poor naive heathens on their borders. Like our good buddy Pope Innocent III said, "so that that barbarous nation may by your efforts become endowed with a more pleasing way of life". Yikes, dude, It's just Scotland.

So back to Shakespeare. The idea of being "English" or an English national identity was already deeply rooted in English culture, particularly as colonialism and expansion of the British Empire continued under Queen Elizabeth and James I. Many scholars, in fact, debate when, in particular, the national identity of England solidified. There has been, in recent years, research on how England began to really lean into the idea of nationalism after Henry VIII broke ties with Rome, with the thought being that Rome had overshadowed the "true glory" of England with its mean religious tyranny of "maybe don't have six wives, bro", but now that H8 had his own sweet new religion, England would finally get a chance to shine. Regardless of if you fall on that side of the argument, most scholars seem to agree that Shakespeare's work actually did a great deal to at least prove that it existed at the time, if not aid in the cementing in its own right.

So to your question of who would have been unified by such a call to arms-- I suppose, at the end of the day, if you're talking about the figurative audience of "the English who would be unified if Henry V takes this advice", or "the audience at the playhouse listening to grand speeches about England". I feel far more confident answering about the latter, as it ties into the significance of the History plays as a whole to English culture.

Helgerson writes that "both the doubleness of Shakespeare's English History plays and the preponderantly royalist effect that the have had on England's self-understanding are related to the material conditions out of which they arose. Those plays were written for a theatre that was patronized one way or another by everyone from apprentices and countrymen to the King" (244), so in some ways, the very nature of the theatrical event became a unifying event in the culture.

Shakespeare (generally) presents English kings in his histories as sort of a stand-in for ALL of England. When England is led by a strong leader, everyone is happy and things are great. When England is under the rule of someone less than great, things are uneasy and there is unrest, either between foreign nations or civil unrest. This wasn't, however, always historically the case, but Shakespeare uses the national condition as a reflection of rule as a dramaturgical tactic to tell the story.

Because the playhouses were frequented by people from markedly different backgrounds , in some ways, Shakespeare's idealistic views of "this sceptered isle, this earth's majesty" act as a sort of nationalistic propaganda while also subtly pointing out the inherent hypocrisy in "look at all these violent murders our forefathers committed in the interest of power, but ain't England great? We're so much more civilized than those OTHER murdering bastards". It's kind of neat that he got away with it, but I suppose that he could have argued he needed to keep the murders in there to sell tickets.

There are instances in Shakespeare's histories in which civil unrest becomes a major part of the plot, but just like the political reality at the time, Shakespeare is careful to tie the fate of those in unrest to that of the ruler-- because civil unrest could spell disaster for any ruling monarch if it got far enough, it stands to reason that linking the fate and destiny of the common people to some noble ideal of being tied to the fate of the very country itself makes for both a more dramatic ending and a less subversive take on civil war on the part of Shakespeare. Unifying the theatrical audience as a sort of microcosm of the entire population of England is a clever way for Shakespeare to say on the good side of the ruling monarch AND gain the loyalty of playgoers, both common and noble, leaving that day's performance of Henry IV, proud to be Englishmen.

This is, like I said, a rather broad answer to a very broad topic, but I hope it was at least somewhat enlightening. Please feel free to reach out if you have any other questions!

Bibliography/Further Reading:
Bartlet, Robert. Medieval English National Identity. The East Asian Journal of British History 1 (2011), 1-12

Davies, R. R. (1990). Domination and Conquest: The Experience of Ireland, Scotland and Wales, 1100–1300. Cambridge, UK: Cambridge University Press

Fox, Alistair. The English Renaissance : Identity and Representation in Elizabethan England. Blackwell, 1997

Helgerson, Richard. Forms of Nationhood : The Elizabethan Writing of England. Pbk. ed., University of Chicago Press, 1994.

Hoenselaars, A. J. Images of Englishmen and Foreigners in the Drama of Shakespeare and His Contemporaries : A Study of Stage Characters and National Identity in English Renaissance Drama, 1558-1642. Fairleigh Dickinson University Press, 1992

Gerald of Wales (Giraldus Cambrensis), Expugnatio Hibernica 2. 5, ed. A. B. Scott and F. X. Martin, Dublin 1978, 146.

Schwyzer, Philip. Literature, Nationalism, and Memory in Early Modern England and Wales. Cambridge University Press, 2010.

Tudeau-Clayton, Margaret, and Willy Maley. This England, That Shakespeare : New Angles on Englishness and the Bard. Ashgate, 2010.