From Wikipedia: "When the input voltage is negative, there is a negative voltage on the diode, so it works like an open circuit, no current flows through the load, and the output voltage is zero."
When the diode is off, shouldn't V- be equal to Vin if we are assuming infinite open loop gain? In that case, isn't Vout = Vin instead of 0 when the diode is off?
To address some of the comments, open loop gain is the gain when no feedback is used. So even if the diode is off, there is an open loop gain A on the node directly in front of the diode on the output. This of course Doesn't reach Vout, but it's there.
If there is an infinite open loop gain with no feedback, as is the case in the ideal d̶i̶o̶d̶e̶ op amp, then this equation must hold: Vout = A(V+ - V-)
If the open loop gain is infinity, then Vout/A = 0 = V+ - V- --> V+ = V-. If the input resistance is also infinity, then there's no current going into Vout and it should be zero, but it's also Vin by virtue of there being infinite open loop gain.
I've answered my own question below and removed my answer from the question.