If you hook it up correctly and the Voltages match... yes.
As for how long... THAT depends on a number of Factors... How much sun, What quality is the Solar Panel, Placement of the Panel.. the goes on and on... But you CAN charge a Battery...I charged a Battery for quite some time using a Large Solar array.... We were Camping an using the Battery for Lights Radio and Fridges.... So it CAN be done
A load tester is typically a resistance pack with an amp gauge to see how the battery deals with current draw. most automotive parts stores have one to bench test car/motorcycle batteries, which they do at no charge (pardon the pun).
Batteries can have a "surface charge"(voltage) which will supply low current at the specified voltage from a battery with bad cells, but when a real operating load, such as for an electric motor, the charge drops to zip..
If the charger is putting out voltage to the battery while it's plugged in then it would definitely be a battery issue, generally the charging system for electric wheel-chairs has regulatory circuits to prevent over charging so having one plugged in when not in use typically doesn't cause issues.
TAKE BATTERY TO AUTO PARTS STORE FOR A LOAD TEST PROCEDURE, MAKE SURE THE CASE IT IS PUT IN HAS THE DOOR CLOSED FOR YOUR SAFETY, BATTERIES SOMETIMES DO "EXPLODE" AND ACID FLYING IS NOT GOOD. IF BATTERY IS NOT GOOD, I WILL SUGGEST YOUR BUY AN AMPHOROUS SOLAR POWERED BATTERY CHARGING PANEL, THEY COST BOUT 30 BUCKS AND WILL KEEP YOUR BATTERY CHARGED UP OVER LONG PERIODS OF TIME.
Using a battery/volt meter, check if the battery has a full charge. Also check if there is an on/off switch on the back of the solar panel. If the batteries are removable and are standard batteries, charge them using an AC battery charger the first time. If not, you may have a defective solar panel and should return it to the store/manufacturer.