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A werewolf from the MM has the following in its stat block:

Damage Immunities: Bludgeoning, Piercing, And Slashing Damage From Nonmagical Weapons That Aren't Silvered

And as established in the errata, unarmed strikes aren't weapons.

Weapons (p. 149). Unarmed strike doesn’t belong on the Weapons table.

and

Melee Attacks (p. 195). The rule on
unarmed strikes should read as follows:
“Instead of using a weapon to make a
melee weapon attack, you can use an unarmed
strike: a punch, kick, head-butt, or
similar forceful blow (none of which count
as weapons). On a hit, an unarmed strike
deals bludgeoning damage equal to 1 +
your Strength modifier. You are proficient
with your unarmed strikes.”

And confirmed by Jeremy Crawford

Addressing a nuance in the PH errata: the rule lets melee weapon attacks use unarmed strikes, despite those strikes not being weapons.

Does this mean my fighter, who doesn't have a magical or silvered weapon can, instead, punch a werewolf in the face for 4 damage?

This does not work.

Your quote is outdated.

The Monster Manual Errata shows that you no longer should reference weapons. Meaning it doesn't matter what you attack with so long as that attack, not weapon, meets the criteria in the entry.

In this case you can make an attack with your unarmed strike that doesn't count as a weapon, but the fact that it is not considered a weapon is made irrelevant.

Damage Resistances/Immunities. Throughout the book, instances of “nonmagical weapons” in Damage Resistances/Immunities entries have been replaced with “nonmagical attacks.”

So the "corrected" entry would read as follows:

Damage Immunities Bludgeoning, Piercing, and Slashing from Nonmagical Attacks that aren't Silvered

This is in fact what the errataed werewolf entry on D&D Beyond now says.

Unless your PC is a level 6+ monk, fists aren't magical attacks

And your characters hands aren't silver. Thus, the attacks are non-silvered, and non-magical, per the MM requirement on how to damage a werewolf.

Damage Immunities Bludgeoning, Piercing, and Slashing from Nonmagical
Attacks that aren't Silvered

What might work is ... wearing silver rings on the fighter's fingers

The "silvering" of a weapon is left a little vague, albeit it is expensive for a low level character.

You can silver a single weapon or ten pieces of ammunition for 100 gp.
This cost represents not only the price of the silver, but the time
and expertise needed to add silver to the weapon without making it
less effective. (Basic Rules, p. 47)

What is "silvering" at your table? How does it work? Discuss this with your DM.

With that in mind, and with Crawfords SA input considered...

Addressing a nuance in the PH errata: the rule lets melee weapon
attacks use unarmed strikes, despite those strikes not being weapons.

... wearing silver rings on your fingers might count as "silvered" for purposes of attacking a lycanthrope with a fist/punch. Work with your DM; it seems a reasonable approach for "silvering" the attacks with your character's hands.

Aside: I've read any number of short stories where were-creatures are foiled by silver rings, silver guitar strings, silver amulets ... but that's a literary reference, not a D&D 5e rules reference.