No, you really cannot use an AC (alternating current) solid state relay (SSR) to control a DC (direct current) load.
The design of a Solid-state Relay will not allow it to reset if a DC load is used.
Explanation Solid-state Relays for AC load switching use thyristors
and triacs as output elements. By turning Solid-state Relays input ON,
the output can be turned ON, but simply turning the input OFF doesn't
mean that the output will be turned OFF. This is because Solid-state
Relays have a characteristic that it stays turned ON until the load
current becomes zero regardless of the absence of input signals into
Solid-state Relays. The DC current will never become zero, therefore
it will not be reset.
The only real surprise here is that it blinks the output LED at all at low power. I'm guessing there is an internal threshold in place. If you are driving DC loads, you will want a solid state relay designed for that purpose.
For DC control of DC loads there are many options available, depending on your output voltage and current requirements, that may be less expensive. (Here's a rough list sorted by loads, small to big: optoisolators, basic driver circuit with BJT transistor, Darlington circuit, Mosfets, IGBTs or if you need it, DC/DC solid state relays) Reference