However, this is as much a faith as a fact, and a paper in the July Nature Reviews ("Comparative studies of gene expression and the evolution of gene regulation," Gallego Romero et al.) systematically reviews the evidence based on new molecular techniques, and suggests that it's a hard question to answer.
The assumption, based on comparative studies, is that much of the variation in gene expression is due to genetic variation and is heritable.
This finding provided a strong motivation for comparative studies to focus on expression levels as an important intermediate molecular phenotype: one that ultimately determines heritable variation in complex morphological and physiological phenotypes, including traits that evolved under natural selection.Gallego Romero et al. describe the state-of-the-art technologies that have been used in this work, including RNA sequencing (RNA-seq), which has replaced microarray analysis in many instances. RNA-seq allows more precision in estimating gene expression levels, which is important for this work. RNA-seq sequences every copy of mRNA extracted from a given set of cells of some chosen type. The more times you see the same gene's mRNA, the higher the expression level. For microarrays, the concentration of a given gene's message was less easily quantifiable.
Assessing the effects of gene regulation on evolution naturally enough encourages evolutionary biologists to try to identify selective scenarios that might explain variation although, as the authors say, "To do so, it is necessary to distinguish between the environmental and genetic effects on gene regulation as well as to control for a large number of potential sources of variation and error" (which can be environmental or experimental).
To do so, it is also necessary to believe that speciation is always due to natural selection. Gallego Romero et al. do cite previous discussion of whether gene expression variation is always such, and clearly themselves recognize that the answer is not necessarily straightforward nor uniform. Gene expression evolution may be due to selection -- often stabilizing selection, which eliminates the extremes, but sometimes directional, meaning that there would be positive benefits to increased expression -- or it may be neutral, that is there's no measurable effect on fitness when expression varies. But, as the paper also says, "Alternative explanations for gene expression differences between species, such as consistent inter-species differences in environments, are often difficult to exclude, especially in primates."
In other words, a definite 'maybe'!
Another complication includes the possibility that gene expression levels may vary by tissue, which at least one comparative study showed. Indeed, as Gallego Romero et al. suggest, documenting variation in gene expression levels across species is the easy part, whatever tissue you choose to use, so long as it's the same for the different species. But there are many issues in deciding what to look at. Making sense of it is much more complex because of questions of how much is due to genetic variation, regulation, environmental influences, explaining underlying molecular mechanisms and so on.
Although progress has been slow, it is now possible to identify functional elements of DNA from nucleotide sequence analysis. Gallego Romero et al. predict that it will one day be possible to predict gene expression patterns from the sequence of their regulatory elements. However, all the same caveats will continue to be true -- gene expression is affected by environmental and epigenetic (non-sequence related changes in DNA) variables, and these will continue to be unpredictable. The authors predict, though, that the use of stem cells will one day make "a reality detailed mechanistic functional studies of gene expression evolution in primates." Stem cells could be induced to behave like, say, liver or kidney or skin cells. Whether they will express genes in the same way out of tissue context as in it is another question.
This paper raises the question of what gene expression variation can tell us about phenotypic evolution, and points out that with new molecular techniques we can document gene expression levels in more detail than ever before. But what these levels actually represent is another question, since, as the paper points out, there are many variables than affect gene expression levels. And, there are numerous reasons for cross-species changes in gene expression levels, including but not limited to natural selection. Indeed, one can imagine that speciation may precede changes in gene expression levels. Or that expression levels just change due to chance changes in the mechanism that don't affect the organism or its 'fitness'.
And as with most aspects of life, there are going to be multiple explanations for speciation. Gene regulation may explain some, but, e.g., Allen Orr is among the more prominent evolutionary biologists documenting genetic causes of speciation such as mutations that create hybrid sterility, or genes that have no harmful effect within a species but when combined with genes from another species cause sterility.
There is no one way and no way to infer from expression differences what their origin might be.