I recently saw someone set a length of galvanized pipe in a concrete footer, and then bolt a pressure-treated fence post to the metal pipe. The galvanized pipe and footer should last essentially forever, and provide plenty of strength to support the fence/gate. The fence and gate can be replaced as needed. I thought it was a fairly clever approach. You might consider it for your application.
It's perfectly normal to have two gates without a center post, using a ground pin (often called a cane bolt) to hold the less-often used side of the gate stationary. The only downside here is that the gate will twist a bit more if side loads are applied near the top of the gate (e.g. wind, someone pushing on the gate), since the only fixed point is at the ground.
The longer the cane bolt, and the further up the gate it extends, the more support it can provide to the gate. Taking this to the extreme, if you were really concerned about the rigidity of the swinging end of the gate, you could make your own "cane bolt" out of a long section of pipe (perhaps 4 feet), and have it drop into a larger pipe set a foot or so into the concrete. That gate wouldn't go anywhere!
If the hinges are on the same vertical line, then (as noted by @DA01) the hinge should stay wherever you leave it (absent wind, etc). If the hinges aren't vertically aligned, then the gate will want to swing towards a specific point. You can use this to solve your problem.
Imagine that the gate is swinging on a rod, and can go 360° around the rod. If the rod is vertical then you can put the gate anywhere you want and it will stay. However, if the rod is leaning towards a certain direction, then the gate will tend to swing until it is pointing in that direction (because the gate will be lower at that point).
What if you move the gate to the exact other side of the rod from where it wants to go? It will sit there, balanced. If you then move it in either direction it will continue to swing in that direction until it reaches its low point.
So, what you want to do with your gate is to align the hinges so that the hinge line leans away from the the middle of the gate's range (at 45°). That way the gate will tend to swing away from the 45° point, towards the nearer of 0° and 90°.
Edit: When you do this, you may have to readjust the position of the gate on its hinges in order to have the gate properly fit into its opening. But, once you adjust the post side of the hinges to get the right swing, changes to the gate side of the hinges won't affect the swing, so you won't have to readjust anything.
Best Answer
If you don't care about how it looks, a piece of plywood or other rigid material mounted to the gate would bridge the gap. You could attach it to the vertical part of the gate using a variety of types of u-bolts/brackets.
Otherwise the easiest thing is to replace the gate with a wider one. Another alternative is to move the posts to reduce the gap on the existing gate but this requires installing posts and replacing the section of fence.