Yes, for some cases.
Assuming the conditions you've given, mainly this is "a world where all vehicles are automated".
Interleaved single vehicles
For intersecting traffic travelling the same speed, vehicles need to stay 7.4 meters apart. Assuming you have perfect information and control of the cars, the spacing is just a car with enough space in front or behind to allow another car to pass as you described. If the car is 4.5 meters long and you only need one meter diagonally, (since the closest the cars come is at the corner for an orthogonal approach) which means $1\over \sqrt 2$ or $\approx 0.707 $ meters in front and back for buffer. Since the car approaching the passing car is approaching the other side, the width of about 1.5 meters also should be accounted for. So, 4.5 m length, plus $\sqrt 2$ m for the buffers, plus the 1.5 m width is about 7.4 meters total space required per car.
If a car is traveling at 100 km/h, it travels its own length and buffer space in just over a quarter of a second. This means if that car is travelling perpendicular to a stream of traffic, there needs to be at least that much time gap between two of the cars (with all buffers accounted for). If they're travelling at the same speed, then that time in terms of distance is that same as the intersecting car length and its buffer.
The density would then be about 135 cars per kilometer. Around "jam density", which is ten times higher than today's "stable" density for traffic flow.
I'm not accounting for the wind turbulence between vehicles passing each other at high speed, which would be rather confounding to this whole exercise.
Interleaved trains of vehicles
As mentioned by JDługosz in the comments the cars could also be in tightly packed groups, with the same average density. So the density won't change, but vehicle efficiency can be increased. Higher efficiency can be gained by grouping vehicles close together because the drag on rear vehicles is decreased; they ride in the draft of the first. The efficiency won't increase dramatically beyond the already significant boost from the sub-10 meter following distance, but would likely be worth the trade-off. Turbulence between passing would be minimized as it only greatly affects the lead cars.
Also pointed out in the comments, the high density closely packed traffic could also be used up to the intersections where the cars actually accelerate to gain the proper spacing. This will still have the turbulence problems as individual vehicles.
The reason this will not happen anytime soon
The human occupants will freak-the-eff out. Grandpa will instinctively lurch the wheel to avoid hitting one car only to slam into another (this is a joke, clearly manual control would be disabled during such a maneuver). You can test this freak out on yourself, just watch a few "near miss" videos and tell me you don't clench a little even watching a video for the second time when you know what will happen. Now imagine being in the vehicle and not knowing.
More likely, for safety and passenger sanity, high speed roundabouts will be used. This would:
- Minimize car-to-car relative velocities.
- The turbulence problems associated with interleaving high-speed cars will be reduced
- Allow vehicles to do something other than go straight through an
- Require more space.
But this would only be for intersections that:
- Don't also need to allow pedestrians to cross, people will still be
walking around in the future, presumably.
- Don't have any other unautomated vehicles, like bikes, segways, or McFly-hoverboards.
- Restrict access to fully automated vehicles.
- Don't have space restrictions like those inside cities.
One other very cool idea about automated cars is having a form of Uber, where people let their car go drive people around in full auto mode.