2003 honda crv fuel guage below empty when actually full

Could be the sending unit in the tank or the gauge on the dash.
A mechanic would test each component by disconnecting the harness at the tank.

The sending unit in the tank is bad or is not positioned properly.

A loose connection on the sending unit could cause the problem but also moisture in the tank in cold weather could also be a factor in which i would add fuel line antifreeze to a full tank of fuel..........Hope this helps you out.

It could be a bad fuel gauge in the dash unit, but....
Usually it's "The bad".
The fuel level sending wipe arm and windings on the fuel sender wear and send erratic readings.The only cure is a replacement fuel gauge sender.
The check engine light should not be related to the erratic fuel readings.
"The good".
If you have over 200,000kms or 120,000 miles on your van you need a new fuel pump anyway,so have a complete assembly put in which will have a new fuel sender on it.

sounds like fuel pump is good the floater may need replacing or the wire that reads the floater may be loose

Probably needs a new fuel sender unit. They cost about $100. The install may be simple if there is a access door in the floor of your vehicle - otherwise you need to drop the gas tank. Expect to pay $400 to $600 to have a mechanic do it.

If you don't get offered a fix, this is a problem you can learn to live with.

Get used to filling the tank to the brim (that may not be so unusual in your part of the world - certainly is here, the way they tax fuel!) and resetting the trip counter on your dash every time you fill up.

For the first few times, fill up after a couple of hundred miles; note the amount of fuel it takes and the distance you've driven. Divide miles by gallons to get your average fuel consumption (average this over a few goes to get a reliable figure).

Find out from the user manual how much fuel the tank holds (if it gives several figures, the one you want is "useable" fuel). Multiply this capacity by the consumption you calculated earlier, and you'll get a very good estimate of the distance you can go before you run out.

Now all you need to do is to keep an eye on the trip counter, and remember to refuel every time you get to (say) 75% of your theoretical maximum distance.

This method has worked well for me over a number of vehicles, none with a particularly effective fuel gauge.

A.

It sounds like problem with either gage or sender unit in fuel tank.
What warning massage from computer.

you can check to see if your guage is getting electricity to it with a test light first and foremost, and then use a wire to directly give it electricity without it being connected to the harness itself and see if it moves too, if it moves with giving it electricity then it's not the guage itself... then check the wires at the tank to see if the harness is getting electricity there. if it is and the guage isn't moving when you wiggle the harness, drop the tank and replace the assembly as a unit that goes inside the tank that has the float attached that makes the guage work.