"Western psychology purports to be "science". That means that results have to be replicable, testable, and observable."Not to go off on too much of a tangent, but the testability criterion was seen as inadequate and was replaced a while back with falsifiability[1], but even that has its own problems.
String theory's claims are supposedly untestable. Is it not a science?
I'm not a physicist, but I've heard there are cloud chamber experiments for which the only observer is the physicist performing the experiment, and are not reproduceable and yet are published in physics journals all the time.
That's not to mention that while many experiments are in principle reproduceable, in practice they usually don't get reproduced, and even if they were reproduced at one point they're not going to be continuously reproduced over and over again by each researcher (who has not the time, education, budget, or interest to reproduce every experiment they've ever read and not yet read but that is still an accepted part of the overall body of scientific knowledge).
Science, just like pretty much every other human cooperative endeavor, largely runs on trust... trust that what one reads and learns is mostly accurate. As skeptical as they make themselves out to be most scientists just don't run out and try to replicate every experiment under the sun or every a large fraction of them. They just trust the results they read are more or less accurate, or expect someone else to catch them.
Back to Western psychology. Some branches of it (like the Cognitive/Behaviorist branches) do purport to be more "scientific", but others like Freudian, Jungian, Humanistic, and Existential and Transpersonal psychology (not to mention art therapy, dance therapy, drama therapy, somatic therapy, life regression therapy, etc) aren't so big on the "science" bit, and aren't so interested in experiments or the scientific method at all.
[1] - https://en.wikipedia.org/wiki/Falsifiability