Can I....

If I have two 6x9s that can handle 50W RMS, and a 4ohm impedance each, and my amp is:
100x2 @ 4ohm
150x2 @ 2ohm

is it possible to run my speakers safely off this amp? Like hooking both speakers to one channel?

It is possible... Depends on how well you know what you are doing though...

What kind of speakers... are they over rated? I would probably go against doing that, unless you are really good with your gains.

If I have two 6x9s that can handle 50W RMS, and a 4ohm impedance each, and my amp is:
100x2 @ 4ohm
150x2 @ 2ohm

is it possible to run my speakers safely off this amp? Like hooking both speakers to one channel?

what brand speakers?????

If they start distorting then turn it down, it's as simple as that. As long as you have them highpassed below ~100hz you should hit the mechanical limits of the speaker before you hit the thermal limits, which goes back to what I said at the beginning....if they start distorting then turn it down. No matter how much power your amp is capable of delivering, the speakers will start distorting at the same power level, and as long as you turn it down when you hear this happening you won't have any more problems with a 2000x2 amp than you would with a 50x2 amp.

Alpine Type S 6x9 coaxial

follow sr20's advice, should be fine

so it should be alright with one speaker per channel as long as the gain is carefully set, correct?

yes

it should be fine, I have ~250x2 powering a set of 100rms components, and I know a guy who has a 500x2 amp on a set of 150rms components. Just don't get giddy with the volume knob and you should be fine, and remember if you hear them distorting, turn it down.