If I have a year in which my capital gains are greater than my employment income and the capital gains, when combined with my employment income, would put me in a higher tax bracket than my normal income alone - will I owe additional taxes on my employment income that was not withheld because the new tax bracket was unexpected?
No. Your taxes on your job will not be affected. However, you will have to pay taxes on your investment income, and that may be taxed at a higher rate than you were expecting.
Think of it this way: each dollar you earn has a certain tax rate associated with it, so you give a percentage of that dollar to the government. As the number of dollars you earn goes up, so does the percentage taken from each additional dollar. Thinking this way, wages get counted first, then investment income. As a result, investment income doesn't affect the tax rate on your wages. But your wages do affect the tax rate on your investment income.
Check the tax worksheet on page 15 of the instructions for Schedule D. This is where you will compute your tax. It's complex, but will give you the right answer.
Note that the additional investment income may possibly reduce or preclude various tax benefits like the child tax credit.
When you file your Form 1040 you compute a total tax. Your question about tax on employment income vs tax on capital gains is a little misguided. It is likely that your withholding will not cover your tax obligation though, assuming that you had the withholding set based on your employment income only and you didn't pay any estimated tax.
This isn't necessarily a problem with the IRS right now, but you should make sure that you're planning to have enough cash on hand to pay the bill in April.
Although it's possible that you'll have an underpayment penalty, in the case that you described it seems likely that you'll hit one of the exceptions and not have a penalty. It wouldn't hurt to do your taxes early though just to be sure and to help plan for the payment. Even if you finish the paperwork this week, you can hold the return and the payment until April 15 if it benefits you. Filing by mail on paper would also buy you a few days relative to e-file if you're cutting it very close. (Don't cut it that close if you don't have to!)