It's amazing how few people have actually addressed your question.
You're right to point out that many high-income (and thanks for not conflating "high-income" with "wealthy") taxpayers earn most of their income from capital gains (and interest and dividends). But Romney's plan does not propose eliminating capital gains tax for high-income individuals -- that's only for taxpayers with AGI below $200,000 (citation here). It does keep in place the lower capital gains tax rates that were part of the 2001-2003 Bush tax cuts -- the rationale being that lower rates on money earned from investment *should* lead to more investment and growth in the overall economy.
Broadening the base means things like capping or eliminating the number of deductions that people can claim on their tax returns. High-income people would pay taxes on a greater portion of their overall income. Romney has refused to commit to a specific proposal on this part, so we're left to speculate. Harvey Rosen has an interesting analysis of how it *could* be mathematically possible to "broaden the base" enough to offset the lower rates -- even with very little additional growth generated via the lower rates. In a nutshell:
You can argue that a Romney administration would never go that far in eliminating deductions and loopholes, but that's different from "mathematically impossible". The Rosen paper puts all of this in a chart so you can compare the effects of different assumptions.
There's a pretty comprehensive round-up of people making the case that Romney plan could work here. (Yes, it's from a libertarian perspective. No, I'm not an economist or a libertarian myself.)
Any program which runs right is obsolete.