You can pick up a cheap water pump that you can use to remove the water. If you have a Harbor Freight in your area, they have a good selection of cheap pumps. If your drain is metal, you can use a brush torch with a 15 pound propane tank to help remove the ice. Use it to help break up the ice. It will probably not be efficient to melt all of the ice. Warm water may help break up the frozen water in the drain pipe.
You don't need to concrete the whole thing.
You do need to shape it correctly. Even in arid areas (seems likely from the picture), water is the major thing that destroys poorly built roads - when it does rain, the water flows down the road and moves material - unless the road is shaped to divert water off to the side in a short distance, so that there is never so much water collectively running on the road that it moves the gravel. In a first step this might consist of adding water bars every once in a while to divert water to the side, but a complete solution consists of putting crown on the road, so that rather than having two wheel ruts (which become water channels, and then become deeper ruts) you have a slightly mounded profile and all water runs off to the sides (and, if need be, you have adequate ditches to handle the water once it is off the side of the road.)
Depending on finances and inclination, you can do this piecemeal over time for little cash with shovels and rakes and implements of destruction, or you can hire a contractor with a road grader. If the contractor also has a rubber-tired vibratory roller and knows how to use it, so much the better. You may want to remove (or break into smaller chunks) the random chunks of concrete, unless you are adding sufficient material to bury them. You will occasionally need to re-rake as your tires move material, but it should not be frequent or major if you drive calmly and are not spinning your tires. If you allow tire ruts to remain, and it rains, you'll have worse ruts after it rains. An old bedspring or section of chainlink fence can also be used as a maintenance drag (vehicle-pulled rake-equivalent) to help keep things where they should be.
If your gravel is 1" down to fines, it should work. If it's 1" stones with no fines, it won't pack well - good for drainage, not so good as a road surface. A gravel/stone supplier should be able to get you a few loads of "road base" which will have enough fines to pack.
If you work on it piecemeal, work from the top down, so each section you complete will stay put.
New townhome has a steep driveway that leads into a ramp up into the garage. Cars are bottoming out as the wheels go into the dip. Looking for advice on the best way to fill or bridge this gap in order to get cars into the garage. 
Best Answer
It looks like the dip is there for drainage, so you can't fill it in - otherwise water from the street would flood your garage every time it rained!
My suggestion would be to get hold of a set of "bridging ladders", as used by off-roaders to bridge gullies - they look very similar to the grids covering the drain in your photo, but thicker, wider and stronger to hold the weight of a car. You could probably bolt them down if necessary to stop them moving, or just store them loose in the garage, depending on how often you need to get the car through.