What is the correct way to pronounce 'router'?

Merriam-Webster lists both ˈrüt and ˈrau̇t as possible pronunciations for route but only ˈrau̇-tər for router.

Is it really wrong to pronounce router as 'rüter ?

There are two different kinds of things called a "router", with two different pronunciations, originating from two different verbs "route" and "rout". The confusing part of this is that the two pronunciations overlap the two different things.

A router as above (computer thing, from verb "route"), or anything else which routes something, is "rooter" or "rowter" depending on how you pronounce "route" (US English has both "root" and "rowt", British English has only "root")

A router (woodworking tool, from verb "rout", an electrical one is shown above) is "rowter", never "rooter", in both the US and the UK and other countries. The dictionary entry may be referring to the wood tool only.

Source: Oxford Online Dictionary & Cambridge Dictionary

The New Oxford American Dictionary (NOAD) reports that router has two different meanings.

router 1 /ˈraʊdər/
noun
A power tool with a shaped cutter, used in carpentry for making grooves for joints, decorative moldings, etc.

router 2 /ˈraʊdər/
noun
A device that forwards data packets to the appropriate parts of a computer network.

In both the cases, the pronunciation is the same.

As comparison, the pronunciation of route (as reported from the NOAD) is /rut/, /raʊt/; the pronunciation of rout is /raʊt/.

In British English, the word is pronounced /ˈraʊtə/ when it has the first meaning, and /ˈruːtə/ in the second case.

Where I live both things are called "rowter" even though we say "root" 66, so if they lived in the U.S. That would be standard. If they said "rooter" it might sound like a thing used to clean drains.

A router (rooter) routes. A router (rauter) routs.

To my ear, the American pronunciation is really grating.