YOu can't have a non fire protected penetration into your home from the garage. That's the first reason you can't run a drain line from the garage to your existing sink. The second reason is that drain pipes are sized to carry the loads they are designed for. They're called DFUs. Adding in a sink to the existing drain will overload the capacity of that drain. The third reason you can't do it the way you want is that I don't see you mentioning venting that drain in any way. Drains don't drain without proper venting.

The correct way to do this is to pull the permit and jackhammer your garage/house slab until you uncover your house's main drain line at which you have a professional attach a drain correctly to and correctly vent that through the garage ceiling, making sure all is properly up to fire codes as well.

In this instance I am assuming the term "unit" is referring to a dwelling unit above the garage such as a garage apartment.

The fact that the existing line from that unit passing through the ceiling into the garage demonstrates that their are provisions for passing a line through a fire rated wall or ceiling into a garage space.

If that is in fact a dwelling unit in the space above it then stands that it must have a minimum of one water closet. Understanding that the minimum line size serving a water closet is 3" it is highly unlikely that there are enough fixtures in that living unit to exceed the DFU load of a 3" line.

The only problem I see is that you may not connect an unvented fixture arm to a stack below a water closet. You would be required to install a vent on the horizontal line from the sink to the stack. Once the vent is installed the line from the sink to the vent location is defined as the fixture arm. The remaining horizontal line from the vent location to the stack is defined as a vented branch.

If you are under the IRC you could install and AAV (air admittance valve) as the vent. The UPC will permit a maximum of one AAV per structure with the expressed written consent of the local AHJ, but in most cases they will only permit it for an island sink, and even then, in many cases they prefer an island loop vent.

If you have to install a separate vent line the line would need to either go through the roof or it must rise to an elevation at least 6" higher than the flood level rim of the highest fixture served by that stack before it could be tied into the stack.

Thank you for the information. Obviously there some things I didn't consider here, like the fire walls. Yes, by "unit" I do mean my condo above the garage. Each building contains four condos. The buildings are situated such that the private garages are on the ground floor. All garages are separated with walls between them, have separate (private) garage doors, and have no direct access to the condo above them. Above the garages are one story condos. Below my condo is my garage and my upstairs neighbors garage. Above my condo is another condo. So building is three stories, garages on first floor, two condos above garages on second floor, and two condos on the third floor.

The plumbing is a bit of a mess. Everyone has a separate water heater in the garage, this is where I would get my hot and cold supply. I do have have supply and drainage lines that penetrate my garage walls and ceiling. Best I can tell, the builder basically split the buildings drainage in half, so that the water from my condo's master bath, kitchen, and laundry and my upstairs neighbors master bath, kitchen, and laundry drain straight down through pipes in my neighbors garage. And the water from my second bath and my upstairs neighbors second bath drain straight down through pipes in my garage. Hope all that makes sense - I'll post a pic of this when I get home. So, with that said, and assuming that a water closet is a reference to a toilet, on this waste line I am referring to, there would be two toilets, two wash basins, and two shower/tubs (half being in my condo and half in my upstairs neighbors condo).

With all that said, as well as what I gather from the above posts, supply isn't an issue, but possibly waste is. The purpose of the sink would be occasional hand washing, so exceeding the line shouldn't be a concern. The only concern would then be the vent if I'm understanding correctly. So because this sink would be installed at a level lower than the lowest toilet, it would need to be vented separately? I would have thought that taping into an existing waste line that is vented would suffice. Does this serve a practical purpose, or is this so that it is up to code?

It serves a practial purpose and is up to code. But it shouldn't be that big of a deal if you can use the AAV.