I'm slightly at a loss as to how to start this answer, because ... kind of, yes? A little?

The French claim

I'm going to be a bit lazy and quote an earlier answer of mine describing the situation:

The crown of France passed directly from father to son in the House of Capet until we get to Philippe IV (1268-1314), who was succeeded by a son (Louis X), then an infant grandson who died soon after birth (Jean I), then another of his sons (Philippe V), and then another. This last son of Philippe IV, Charles IV, had only one child living - and she was a girl: Marie. His wife was pregnant at the time of his death, but the child turned out to be another girl, Blanche.

Unlike certain other countries, Capetian France never had a king before Louis X who didn't have a son to pass the crown to. His uncle Philippe was made regent until Louis's pregnant wife gave birth to Jean, and then decided that he should inherit the throne instead of Jean's older sister, Jeanne. He had the Estates-General declare that there was insufficient precedence for a woman to inherit, and Jeanne was allowed only the throne of Navarre. There was no problem with Charles's accession after Philippe's, but with no more Capet brothers to succeed him, it was necessary to take stronger steps. It was declared, on the basis of the old "Salic Law" [...] that women definitely had no right to a portion of the kingdom, no ability to inherit it or to pass it on to their children, which excluded Marie and Blanche.

Instead, Charles IV was succeeded by Philippe VI.

Shakespeare's riposte

To summarize the Archbishop of Canterbury's speech from Act I, scene ii, Charlemagne conquered "the land Salique" (around Dresden) and installed some of his Franks there. These vassals married local women but felt they were sexually incontinent, and so they created a law that no women could inherit in the Salic land - the implication is that they couldn't inherit positions or property.

However, the French also claimed that the law was created by King Pharamond, the legendary first king of the Franks, who would have existed over four centuries before Charlemagne took Salia(?).

The Archbishop then points out that Pepin the Short, Charlemagne's father, deposed Childeric III and became King of the Franks based on his descent from Blithild, a daughter of Clothar I many generations back. Likewise, Hugh Capet - the progenitor of the powerful Capetian dynasty that ruled France until [all the stuff that kicked off the Hundred Years' War] happened - usurped his own predecessor while claiming descent from a daughter of Charlemagne, and Louis X also made sure that his grandmother could be said to descend from the daughter of the king Hugh deposed. Since they all "hold in right and title of the female", this is further proof that the Franks always knew that Salic Law referred to that specific region of Germany and was not to be applied to France.

The truth (as best we can tell)

Shakespeare's Archbishop is absolutely correct that the French didn't seem to have felt that Salic Law (which was actually a full law code for the Franks started by King Clovis I (466-511)) applied to them until Philippe VI. They actually didn't even mention Salic Law as the basis for this change to the line of succession immediately. According to a contemporary chronicler, keeping the English Edward III out of the line of succession was an explicit part of the discussion the French lawyers had in deciding what the law said about who should be the next monarch.

To add to this, the Salic Law did not bar women from all forms of inheritance. According to Robert Bartlett in Blood Royal: Dynastic Politics in Medieval Europe, a clause in the Salic law code that barred women from inheriting "a particular kind of private property" was pulled out of context by the scholar Jean de Montreuil in 1413, and the words "in regno" inserted. This made the clause read that women had "no share in the kingdom", and led to the conflation of Salic Law specifically with this principle. (To be clear, they were invoking Salic Law before this; 1413 is just when they started to claim that Salic Law was explicit on the point.)

The contemporary English argument

I don't know what exactly the average Englishman's belief was, but Edward's lawyers made cogent arguments that had to be responded to by the French. And they made the point that Salic Law was being invoked (when it eventually was invoked) to give the impression that the principle of women not being able to pass a claim to the throne to their sons had long standing in France, but that the principle had in fact been invented in 1328. At the same time, they agreed that it prevented women from being queens in their own right: they just disagreed that men could not inherit the throne through women, because all the law said was that a woman couldn't succeed and made no mention of her son.

There doesn't seem to have been a real thread in the English arguments, at least in Edward's time, of this portion of Salic Law applying only to German women in the Dresden area. That would have been counterproductive, since they did want to disinherit the female heirs between Edward and the French throne. It's possible that it became more common by the Elizabethan era, at which point it was really all moot, but I don't know about that. However, Shakespeare's Archbishop's overall message, that French claims about Salic Law were trumped up and contradictory, was correct.