The terms "right wing" and "left wing" are highly variable depending on which country one is talking about, so I don't think there's any universal definition whereby all capitalists are necessarily right-wing. For an example focused on the social (rather than economic) aspect of the terms, merely supporting the freedom to have premarital sex, the freedom to be openly irreligious, or even gender equality is strongly left-wing relative to Saudi Arabia's social and political spectrum, even though many people who have these beliefs can still be socially conservative relative to some other countries' contexts.
There's also a difference between social and fiscal spectrums, of course: One could be socially liberal and fiscally right-wing or socially conservative and fiscally left-wing. Calling someone's views "right-wing" or "left-wing" wholesale without clarifying which aspect thereof is being described leaves a lot of room for ambiguity. Islamists are so socially conservative that many of them endorse lashing for premarital sex and capital punishment for homosexual sex, but many Islamist groups actually tend to be economically leftist in a few key ways, such as their staunch support for a strong welfare state and belief in a greater social responsibility for wealthy people.
It is true that in some political contexts, capitalism is typically either near the center or to the right of center on the fiscal spectrum, although it's also true that, say, libertarianism and laissez-faire capitalism are a lot farther to the right than social democracy, even though the latter still retains capitalism and doesn't eliminate or replace it like solidly left-wing economic ideologies tend to propose.
Personally, I don't care about labels—much less binary, simplified, and highly generalized labels—as much as I do people's actual values and beliefs. Some classifications are sometimes quite vague, relative, and variable, as I outlined above, but more importantly, asking people questions and letting them clarify their views seems to me a much more reliable way of understanding others than pigeonholing them as X or Y and trying to force them into a label.