Short answer: No.
Long answer: One may influence the other in some cases, but not in a deterministic way that would enable accurate prediction. Read on - the following is basically my thoughts on the hypothetical situation where, upon learning that someone is "trans", a person might assume they can deduce that person's sexual orientation. Or, in other words, the "only gay men wear dresses" fallacy.
It is obvious to me (more so than it apparently is to some others) that sex and gender are two very different things (and that orientation is something else altogether). One of them has more to do with your anatomy and physiology, and the other has more to do with your psychology and personality. That there are two obvious sexes and two obvious genders (at least - gender is a far more nebulous concept than sex), and that the majority of the population goes through life never seriously questioning the traditional pairing of these (i.e., men are masculine, and women are feminine), explains why those who experience life otherwise are a commonly misunderstood (or downright ignored) minority. But it doesn't diminish the value of their experiences, the legitimacy of their identity, or their claim to the same rights and freedoms the rest of us have.
Now, I wouldn't go so far as to say that sexual orientation and gender identity are completely independent of one another. I believe they can have strong influences on each other. But the fact is that different people ultimately come to different conclusions (about what gender they identify with, and what "sex" they are attracted to), and the determination of one isn't enough to predict the other. Just as some males can be masculine and others can be feminine (and some females can be feminine while others are masculine), a person who identifies as the male gender may be attracted to either males or females (or both, or something else altogether), and the same can be said of a person who identifies as the female gender. The antiquated notion that if a man identifies as female, for example, he must be sexually attracted to men, is blatantly heteronormative - denying the existence of females who are attracted to other females - and gender normative - assuming that no "red-blooded" male who is attracted to females could have stereotypically feminine interests.
I can't, personally, say that my sexual orientation hasn't influenced my gender identity, in the sense that I am fascinated by females so much, I began to think I'd enjoy being one; and after experimenting, discovered that it fit me so much better than a male identity ever has. In a different case, a person with a sexual attraction to males (who, as a male, would be labeled "gay"), might begin to identify more with females from something of a vaguely heteronormative (which is NOT to say homophobic) perspective, in the sense of wanting to be the "female" to a male's desire. To be true, this is not how all (or probably even most) gay men feel. And the sex (or sexes) that one is attracted to does not necessarily impact the gender they identify with - they are two different feelings, two different "senses". But it's just one of a variety of fabulous possibilities.
Which is why I can't say that sexual orientation and gender identity are completely independent of one another, in all cases. But at the same time, it should be kept in mind that they are not determinant predictors of one another, either. In logical terms, if A may influence B or C, then it could be said that there is a connection between them, but knowing that something is A is not sufficient to predict whether it will align with B or C - as both possibilities exist. You cannot look at a woman and know with a glance whether she is straight or lesbian (or something else). It is no less true that you cannot look at a man wearing a dress and know instantly who or what he is sexually attracted to.