SSRs are Mity Fine. No problem on driving the SSR right off a pin… which you already proved using a bulb as SSR load.
An old fashion tungsten bulb is a tame load, you might wanna look at one other thing before you try to kerchunk the solenoid. A solenoid has inductive “kick” and has the same issue with “punching through” the SSR as a conventional relay coil has in “punching through” a microcontroller output pin or transistor. See note 3 of your data sheet.
3) Inductive loads should be diode suppressed
So although you can safely connect the SSR control pins to Arduino, you may need an RC snubber or diode to protect the SSR from the solenoid. We assume its an AC circuit you cant simply use a regular rectifier diode across the solenoid. You need a TVS diode or a MOV or something like that.
The SSR may take the abuse for a while, they are pretty rugged. Even after it punches thru your SSR may be stuck ON or OFF, but the SSR isolation should still prevent any badness from getting back into your Arduino.
Consider the failure mode of the suppression part in your design.
Gas Discharge Tubes usually fail OPEN. This causes no obvious malfunction, but the loss of protection will cause frequent failures of the (formerly) protected component.
Zener diodes, MOVs and solid state suppression generally fail SHORTED. So at end of life the fuse or breaker trips until the part is replaced. This is usually preferred. (you DO have a fuse or breaker DONT YOU ?)
for 120 Vac mains I have used P6KE180CA see it here
if youre too excited and dont wanna stop playing solenoid, just hack open a cheap surge strip and steal the MOV from it.
Good luck with your project!