Can a riverboat tack upriver?

No, because the relative directions of forces and travel are wrong

Consider your diagram. In order to change the frame of reference, the current would have to be pushing the boat left to right, in the opposite direction to the arrows above the green dots. The result of this would be an "apparent wind" creating the force in the direction shown.

Unfortunately the direction is (at least partially) in the same direction as the current. The sail will allow you to cross the current, but it can never produce a force which would act against the direction of current flow.

There was a wonderful article in the American Institute of Aeronautics and Astronautics (AIAA) journal (80s? maybe 70s?) envisioning a sailboat that was nothing but an airfoil (the sail) connected to a water foil (the keel). Based on the density ratio, the airfoil should be 80 times larger (IIRC) than the water foil. It then imagined people who live on the ground and ride on the water foil and people who live in the air and ride on the air foil. Just as ground dwellers wouldn't think of sailing upwind, air dwellers wouldn't think of sailing upwater. The natural origin of coordinates is centered between an air point and a water point. You can easily get half the wind speed in either direction by feathering one foil and turning the other perpendicular. The real question is can you do better by reaching. Experience says yes, because sailboats can make distance to weather in a ground fixed coordinate system. Your question is just the reverse, so properly constructed boats should be able to make distance to water in a ground fixed coordinate system.

Shame on the physics SE for closing it.

No tacking required, although it is optional. The vessel is going directly down wind, not up wind.

There is a factor called the speed ratio (SR), which is the boat speed divided by the true wind speed (TWS). TWS is the scalar difference between the wind velocity and water water velocity. And there is the associated true wind angle (TWA), which is the angle of the wind with respect to the desired vessel course. If the river is flowing and the air is stationary, then you need a speed ratio greater than one going directly down wind in order to go upriver, or a speed ratio of more than 1.414 tacking down wind at 135 degrees. There aren't many boats that can do that. The last generation of America's Cup foilers could. And to answer your question, this was a longstanding challenge and I believe it was first validated by a custom windsurfer with about four guys on it maybe twenty years ago. That's all from memory, though. There are also wind prop/water turbine boats where the water prop drives the wind turbine. These can travel downwind faster than the wind.

The key to all of these DDWFTTW devices is that the speed through the water (or over the ground) is faster than the speed through the air, so you generate power from the faster stream, which is the water, and deliver it to the slower stream, which is the air. And you can generate a net thrust doing that even if the system is not terribly efficient. Now if the net thrust exceeds the vessel's drag at river flow speed, you can make headway upstream.

See here for a discussion of what amounts to the same problem - https://www.boatdesign.net/threads/ddwfttw-directly-downwind-faster-than-the-wind.25527/

Blackbird - https://en.wikipedia.org/wiki/Blackbird_(land_yacht)

Some proof of concept videos -

http://www.youtube.com/watch?v=aJpdWHFqHm0

https://www.youtube.com/watch?v=xHsXcHoJu-A

https://www.youtube.com/watch?v=Ava7C62TSZ8

https://www.youtube.com/watch?v=DPvGTjmn9y0

Probably yes. To a sailboat, there's no difference between floating in a river surrounded by stagnant air, and floating in stagnant water surrounded by moving air (wind). The boat will "feel" a relative wind in both cases. The only question is whether or not a sailboat could sail upstream faster than the water is flowing downstream.

The relevant nautical term here is velocity made good (VMG), which is the velocity at which a boat can sail directly upwind by tacking back and forth. High-performance racing yachts routinely achieve VMG in excess of the actual wind speed, sometimes by as much as a factor of 2. So, at least in theory, a modern yacht would be able to sail up-river only using the river current.

Two caveats however: These feats of speed are typically achieved in windy conditions, so your river would probably have to be flowing pretty fast for this to work. Also, as was pointed out in the comments, the water velocity in a river is not smooth and uniform; there are all kinds of eddies and edge effects that might work against you.

River water flow creates hydrodynamic pressure which is equal to

$$ P_{hydrodynamic}=1/2 rho v^2$$

If we can tap this pressure to do work it is possible to use it to move the boat upstream.

One practical solution would be to have a paddle wheel like those used on riverboats turn a winch winding a cable that is anchored to the shore upstream at the desired destination.