I'm reading through some radar books and I'm trying to find out if there's an easy method or equation to recalculate radar detection range given only the original radar detection range, original probability of detection (pd) and probability of false alarm (pfa), and a new pd and pfa. Is this possible or do I need to consider more input variables?
No, you need lots more. Or you can assume, whether right or not it will depend on the assumptions and the real conditions.
The most benign conditions is White Gaussian noise, and radar propagation in vacuum. That what I do when I know almost nothing.
If you assume a Gaussian or a Raleigh or Riccian pdf you can related Pd and Pfa to SNR, if you can assume the Raleigh or Riccian parameters. So you know the SNR for the first and second cases, and you know the range of the first. Now you need to figure out how Signal power would vary as function of distance, and then you can get the range for your second case, if you again assume that it's the same radar, with the same noise and internals, and assume you are trying to detect the same target in both cases (same angles, same cross section and fluctuation statistics). So those assumptions might be ok.
Unfortunately, the signal power will depend on distance depending on the environment. In vacuum it's $1/r^4$, because of the two way path. But if there's any kind of multipath effect from the ground it could be worse. If not ground to ground radar you can assume you can ignore the multipath, otherwise not.
So, it depends on the situations. But if you can safely assume most of those, you got it.
On the other hand, buy a good radar book and it'll tell you much better what some of the parameters might be and you can see some of the graphs.