Here's the deal. I think America has, for now, settled on roughly the following divisions in race: White, Black, Latino, Native American, and Asian. Arab strikes me as being seen more as an ethnicity than a race, but that may be changing with the whole "clash of civilizations" rhetoric surrounding the war on terrorism (there's some really interesting literature about how Arabs "won their Whiteness" in legal cases back when America still made serious racial distinctions in its immigration laws). But aside from Arabs, Indians are the group that really gets left out. But it's interesting as to why.
In terms of racial stereotyping, I feel like Indians should be grouped with their geographic compatriots, (East) Asians. Aside from the fact that Indians are from, well, Asia, they are viewed in much the same way by Whites as Asians are. The stereotypes are predominantly (not completely) positive. They are seen as academically advanced, even nerdish. Mockery tends to be based around accent (the most prominent uniquely Indian negative stereotype, the Apu-type shopkeeper, is based heavily around this). Sometimes they are seen as competitive threats (taking our jobs), but are not usually considered inferior (sort of like Jews). They are not seen as particularly criminal or violent. Interracial relationships between Whites and Indians, like White/Asian pairs, are relatively accepted compared to other cross-racial mixes.
Yet, I don't think that when most people talk about Asians as a race, they include Indian-Americans. Why is this? In America, race is inextricably linked to color. That's why the binary races are called "White" and "Black". Archaically, you'd see Native Americans called the "red" race ("Redskins"), Asians "yellow", Indians "brown", and Latinos "tan." Today most of that has fallen by the wayside except White/Black. But the mental linkage remains, and let's face it--east Asians and Indians are of a different color. Hence, I think we have a lot of trouble placing them as part of the same race.* It'd ruin our neat framework that unites race and color.
Ideally, the tension Indians bring out between color-centric and characteristic-centric manners of racial categorization would help reveal some of the instability of racial categories, and would lead to some more critical dialogue about what it means to be part of a "race". Unfortunately, America as a nation seems to have a collective inability to deal with complexity ("John Kerry is too nuanced for me!"), so that's out of the question. What we get instead is Indians being dropped out the conversations entirely, so we don't have to deal with the issues they raise to our basic conceptual schema. Similarly, non-Arab Muslim states are either simply grouped as the same as their Arab neighbors (Iran, Pakistan), or relatively ignored when talking about how Muslim governments do or do not perform (Turkey, Indonesia). Arab=Muslim=Simple. Arabs being of many religions, and Muslims being of many ethnicities, is not simple. Intersectionality questions, ditto ("Black AND Woman? One at a time!").
So to answer your concern, PG, the reason Indians are absent from American racial discourse is because they ruin our neat conceptual frame. Stop complexifying our simple little world.
*To be clear, I don't think we actually follow any of our racial categorization procedures to their logical conclusion, because it would expose the concept of race as the absurdist fantasy that it is. Color doesn't fit because it's nearly impossible to distinguish the hue of Latinos from that of Southern Europeans. Stereotypes don't work because they don't actually track groups neatly like we pretend that they do. Biology doesn't work because ethnically related groups are spread out too far to keep our idea of "distinct" races--
UPDATE: I'm withdrawing the Finn/Korea connection claim because I've been unable to find significant corroberating evidence beyond my original source (my tour guide when I was in Finland). I have noticed some similarities in physical appearance, and have since read some evidence for the linguistic connection, but it isn't strong enough to base a claim on.