Short answer

Not by itself. A private window is built to forget: history, cookies and site data are thrown away when you close it. A fingerprint is not stored on your device at all, so there is nothing for the window to forget (EFF). What can help is a browser that adds anti-fingerprinting measures in private mode, and that differs from one browser to another.

Chrome Incognito

Google's help page lists what Incognito does and does not do (Google):

Removes at the end: site data, cookies and the record of sites you visited.
Blocks: third-party cookies, by default.
Does not hide you from: the websites you visit, Google sites included, or whoever runs the network (a school, employer or internet provider).
Keeps: bookmarks and downloads.

The page does not mention fingerprinting or IP addresses. Nothing on it says Incognito changes the screen size, time zone, fonts or other details a site can combine.

Firefox Private Browsing

Mozilla goes further in private windows: both its Known Fingerprinters and Suspected Fingerprinters protections are switched on there, while normal windows have only Known Fingerprinters on by default (Mozilla). Suspected Fingerprinters is the one that adds canvas noise and rounds hardware values. Details on our Firefox page.

Safari's private mode has its own set of protections; see Safari Private Browsing.

Side by side

Private window in	Forgets cookies and history	Extra anti-fingerprinting
Chrome	Yes (Google)	Not mentioned by Google
Firefox	Yes	Yes, both protections on (Mozilla)

See the difference yourself

Open the browser check in a normal window and note the values.
Open it again in a private window of the same browser.
Compare. In Chrome you will likely see the same screen, time zone and hardware values. In Firefox some values, such as processor cores, may be rounded.

The check runs entirely on your device, so trying it in both windows sends nothing anywhere.

The private-window myth, in one table

Belief	What the sources say
"Sites cannot recognise me in a private window"	Sites can still observe your activity (Google)
"My employer cannot see it"	Network owners may still see activity (Google)
"Clearing everything resets my fingerprint"	A fingerprint is not stored, so it cannot be cleared (EFF)

Outside this page: Private modes in mobile apps and in browsers whose help pages we have not read today.