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Sure it can. There should not be a completely random pattern of mutations in organisms if mutation and natural selection is the mechanism of speciation.

I don't agree. See below.

If mutation and natural selection is the mechanism, you would expect to see the same mutations in different species.

Actually, I wouldn't expect to see the same mutations, even between individuals within a single species. Were I to keep seeing the same mutations over many unrelated individuals, then I would begin looking for some commmon mechanism that was causing those mutations to occur.

The idea of natural selection is that it "rewards" certain mutations, and "punishes" others. Many, perhaps most, mutations would be "indifferent." Both the "good" and indifferent traits would tend to be carried forward. But the underlying mutations are assumed to occur randomly.

This is an over-simplistic description of natural selection, as it tends to focus on single mutations in isolation. That is not how nature really works. The mutations themselves are random and isolated, and in most cases I suspect that a particular mutation is indifferent, or maybe even "bad," except when accompanied by other, complementary mutations. It is the set of mutations, working together, that would confer a survival advantage.

Another danger in the usual top-level description of natural selection is that we tend to focus on end results, and thus too much on "what happens next," even though the mutations, being random, are selected-for in the here-and-now, and CANNOT have any view of what future mutations might bring.

In practice, we take an end result (e.g., a flying animal), and then begin constructing a likely chain of mutations that could have produced the characteristics of the flying beasty. Which is to say: we design (or perhaps more accurately, reverse engineer) a process of natural selection that could have produced the animal we see flying outside. But of course, what can be reverse-engineered, could also have been engineered in the first place. So again, there is nothing in the idea of mutation/selection that necessarily precludes the presence of an intelligent designer in the loop. Selective breeding and genetic engineering are obvious example that this can and does occur.

If you observed completely different mutations in different species, then mutation and natural selection is falsified.

Wrong, for a couple of reasons. First, that statement requires us to assume that identical environments (if such could be found) always select for the same mutations, in the same order, and in the same way. And from that, you have to assume that there is only one possible "solution," to which natural selection is always driven. The mere presence of those "different species" (some of which are very different, indeed) is enough to show that neither assumption is true, much less required.

I'm not at all surprised when I see the results of extremely different sets of mutations -- insects vs. mammals, for example.

(Note the converse is not true, ie. if you observe the same mutations in different species it is not necessarily the case that mutation and natural selection must be true. It does lend support to that model, however.)

You're correct: there is no logical requirement for similar end effects to have arisen from "the same" mutations.

On a logical note, you don't seem to have constructed a true converse here. Your original statement was ["different mutations" -> "natural selection is false."] The converse of that is ["natural selection is false" -> "different mutations."]

You've constructed more of a contrapositive (a->b == ~b->~a) and, because the original statement is not logically sound, you find yourself in a position of ambiguity when you form its contrapositive. Thus it's no surprise that you came up with "it's not necessarily the case" for the latter, because it's also "not necessarily the case" for your original premise.