Windshield wiper spindle

A good bet would be you most likely stripped a few of the splines located where the wiper arm mounts to the motor assembly. It's usually under a cap or lid of some sort.
If the wiper arm moves freely you could try to tighten the nut a bit more to secure it...this may work but won't solve the problem.
You'll most likely need to replace whichever part lost the most material (usually the wiper arm).
Good luck!

There is only one motor and under the windshield cowling is an arm that connects the left and right side. The wiper arm sits on a spindle that is splined. Check for stripped or loose spindle or connection on those parts.

Either the linkage is stripped out or broke. I would look at the spindle that the blade attaches to. On the underside of it is a linkage that must have something wrong with it. Please rate my answer so I am rewarded points for it.

The wiper arms are 'pressed' onto the wiper motor spindles, often at times when only one of them stops operating, it is because the splines on the wiper that is no longer operating has become stripped out. and may need to either be replaced or tightened. I would recommend tightening it first, then replacing it if that does not correct it.
You can access the hold down bolt (if it is not exposed) by taking a small screw driver and prying up the bolt cap. The bolt is usually a 13mm, and is more easily tightened by a ratchet and socket of the same size.

At the base of the wiper arm there is likely a plastic cover or cap. When you remove that cap or cover there will be a nut that holds the wiper arm to the post. Align the arm back to it's proper position and tighten the nut. Typically they are aeither a 15 mm or 13 mm nut.

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First remove the complete passenger side wiper check and make sure splines on wiper are not stripped. If this is not the problem It could be a linkage bar from the wiper motor to the wiper off or disconected. Since both wipers share the same motor if one works the other should. Check this by removing both wipers and then "Cawl" (little black piece between the hood and windshield underneath the wipers) Underneath the cawl you should be able to see the windshield wiper motor, and linkage bar to both of the wipers. See if the bar is disconected from the passenger side. If it is than reconnect it. These are the only two possible scenarios.

In view of the answer from jchaffcom I feel that if there is a Hex on the end of the Wiper Arm the Hex Nut needs Tightened. I have a 2004 Silverado and this is what happened to it. Simply tightening the Nut solved my problem. Try it and get back to me with the results whether good or bad.

Fastest Robe

the wiper arm assembly is stripped for the knurled knob that is sits on.  replace the wiper arm with a new one.

I just solved this problem last night with my '05 Impala, and ultimately the solution was what newton5083 stated above.  Yet, it may also be this (I found it online somewhere):
You have bent or broken the "park tab" on the windshield wiper motor bracket. The correct way to fix this is to remove the wiper arms, wiper linkage and wiper motor from the wiper motor bracket, and install a new bracket.
I just bent the tab back into place a couple times for our daughters 2005 Impala but it kept getting bent down again when they didn't take the time to make sure the wipers were not frozen to the windshield, or tried running the wipers to remove heavy snow on the windshield. So yesterday I fabricated a small piece of metal and bolted that to the bracket in the same place the tab stop was.
If someone wants to access this bracket and tab-
1. Use a ½ inch or 13mm socket & ratchet or wrench to remove nuts on wiper arms.
2. They may make a special tool to remove the arms, I used a screwdriver and hammer to tap bottom of the arm to wedge them apart from the spindle.
3. Two small button-looking clips hold the drivers side cowl cover in place under the windshield. These are two-piece clips, pry up the center, than the outer piece.
4. Unclip the washer hose from cowl cover, than pull out plastic cover.
5. There is a smaller plastic cover with on plastic button head holding that cover in place. Gently pry under that clip to remove that cover.
6. Now you can see the wiper trans and bracket. You should see a small tab on that bracket that looks pushed down and bent or possible a square small hole with some torn off metal where the tab broke off.
7. Remove small plastic cover over the wiper transmission and you can see a small spring-loaded arm underneath the wiper transmission assembly.
8. You need to find something to pry that tab back up so when the wipers are running that spring loaded little arm just snaps past the tab running clockwise. It's when you shut off the wipers that the wiper transmission assembly reverses rotation counter clockwise and the small spring loaded arm underneath hits that tab stop, than the wipers will go all the way down and shut off as they should.
Once that tab stop gets bent out of the way the little spring loaded arm has nothing to stop the wiper transmission assembly so it keeps traveling counter clockwise until some other stop shuts the wipers off straight up on the windshield.

There is a revised bracket part # which is supposed to be more durable, GM part #12368639. Here is a copy of the TSB which points out this problem:
Bulletin No.: 05-08-43-003A
Date: March 10, 2005
TECHNICAL
Subject:
Windshield Wipers Park in Up Position (Replace Windshield Wiper Motor Bracket)
Models:
2005 Chevrolet Impala, Monte Carlo
Supercede:
This bulletin is being revised to make a correction to the Parts Information. Please discard Corporate Bulletin Number 05-08-43-003 (Section 08-Body and Accessories).

Condition
Some customers may comment that the windshield wipers park in the out wipe (vertical) position.
Cause
This condition may be caused by a bent park tab on the windshield wiper motor bracket.

Correction
To correct this condition, replace the wiper motor bracket. It will be necessary to transfer the following parts listed below to the new bracket.
^ Wiper transmission
^ Wiper motor crank arm
^ Wiper motor
1. Remove the windshield wiper motor drive system module.
2. Remove the windshield wiper transmission from the windshield wiper motor bracket.
3. Remove the wiper motor crank arm from the windshield wiper motor.
4. Remove the windshield wiper motor from the bracket.
5. Install the windshield wiper motor to the new bracket.
6. Install the wiper motor crank arm to the windshield wiper motor.
7. Install the windshield wiper transmission to the windshield wiper motor bracket.
8. Install the windshield wiper motor drive system module.

A TSB is not a recall, if vehicle is still under warranty the dealer should be repaired for free. If not, you'll have to pay for the repairs. 

thers a bolt where the wiper piviots and it might need tigting