Yes, you can charge each battery independently from separate solar panels as long as each is electrically isolated.
However, it would make more sense to combine the panels and charge the whole 24 V battery as one. Lead-acid batteries are fairly forgiving, especially in that you can charge them with a current-limited voltage supply. A 12 V battery can be held indefinitely at about 13.6 V without harm. It will draw some charge current at that level, but this current won't do long term harm even if the battery is already fully charged.
A power supply that takes the solar panel output and makes the roughly 27.2 V output should work fine. The more cells in series, which in this case is 12 instead of the usual 6 of a "12 V" battery, the more you have to worry about cell imballance. You might therefore want to be a bit conservative with the float voltage and/or make the current limit lower than usual. The easiest way to do that is to size the solar panel such that it simply can't produce enough power to hurt the battery in full sunlight. Then the power supply only needs to try to produce about 27 V, but fold back as necessary limited by the power from the solar panel. This scheme will give up fast charge time, but if that is not a issue it's intrisically safe operation makes things easier.
Taking the question at face value, ALL the answers given are wrong [ :-) ] and your question and a subsequent comment do not match.
You say you have 4 x 12V batteries to make a 48V supply.
Your diagram implies (but does not state) that the panel is 24V rated.
In a comment you say
I don't think I know the PV array's knew voltage. I know it produces up to 18V in full sunlight though. – Wurlitzer Jul 10 '12 at 23:00
An 18V O/C panel is 12V rated, not 24V as per diagram.
Cheap and nasty solution:
Use a relay (or relays) to swap the panel between batteries. Either two positions if 24V panel or 4 positions if 12V panel.
A simple sequencer will be required to control the relays.
Very fast swapping rate reduces relay lifetime, and controller will take a time to settle so "dead time" % is higher.
Very slow swapping rate increases risk of imbalance.
If you swapped every minute or even every 5 minutes the long term differences will be minimal.
A good "real" solution is to use a 48V panel - or 2 x 24V or 4 x 12V panels in series.
Also suitable would be to use a 12V to 48V converter. Some MPPT controllers are available which will both up-convert the input voltage and also optimise loading to maximise panel power output.
Best Answer
You can charge two batteries together, you can charge them either in series or in parallel. Looks to me you'll be charging them in series because that's the way they'll be connected.
But I'd suggest you use a proper charger circuit to do it. You haven't said what kind of batteries they are and that matters as to how they should be charged. The simplest charger I ever saw was a resistor connected to a NiCd battery! All that did was limit the current flowing into the battery so it charged up slowly. But not recommended! You need a way to detect when the battery is full and stop charging it (as overcharging damages the battery).
The other issue you have got is your 24 volt panel. It will may only produce 24 volts when in maximum bright sunshine. If you've got a charger circuit that will use some volts, which generally means the output of the charger is going to be less than 24 volts, which means it probably won't be able to charge the battery fully. (As the battery charges up, typically the voltage increases. You've got 2x 12 volt battery, for a single battery (lead acid type), the terminal voltage can go up up higher than 12V, 12.5, 12.8, and the charger voltage will go up as higher as 13.8V in order to charge it. So chances are you are are not going to be able to charge a 24V battery (2x12v) fully with a 24 volt panel and a charging circuit, unless you start using sophisticated chargers, DC converters which can step up the voltage.
And you might want to stick a diode in series with the panel and what it's feeding power to (the charger, the batteries), as you may find (depends on the panel) in low levels of sunlight, the batteries discharging back through the panel and draining them. Particularly true if you leave the batteries permanently connected to the panel.
Look around for 24Volt solar charge controllers, see whether something like that might be suitable.