An air carrier First Officer reported his Captain seemed to have situational awareness and CRM issues that led to a deviation from their charted arrival.

Narrative

[We were] descending on the EAGUL Arrival into PHX as the pilot not flying. When we were cleared to descend via the EAGUL; maintain 300 until PITCR; then via the arrival speeds; I asked the Captain if he wanted me to set 7000 which is the final altitude for the arrival in a managed descent. The Captain; at one point; decided to descend using selected speed and descent rates. He cleared the 240 altitude by using vertical rate 0; instead of having me reset the altimeter limiter to the 240 and using the old conventional descent method. The arrival has [a restriction to] cross PITCR at 16;000. Inside the point 6.0 prior to PITCR; EAGUL; I think; the Captain was descending still. At 4.5 prior to PITCR; the Captain started to descend below 16;000; in violation of the cross at 16;000. I immediately called out the error yet the Captain continued to descend. He finally leveled off at 15;200; but did not even attempt to aggressively go back to the altitude. The only way I could have made this less of an issue was to have taken the aircraft from the Captain and leveled it off. Considering that the Captain tried to ream me at the gate for attempting to fly his airplane as I tried to assertively get him to level off when it seemed as though he had no situational awareness that would likely have been a not well-accepted maneuver. I did advocate using the conventional method of setting each altitude for the rest of the descent; and we did use that. The Captain's final position at the gate was that I should give him more of a heads up when he is about to make a mistake. Not sure how to do that; as I don't know he is going to violate an altitude crossing restriction until he starts an open descent and actually demonstrates a lack of situational awareness.