I know a Kolmogorov-Smirnov test will tell me if a sample distribution belongs to a normal distribution or not
No, it can never tell you a sample was drawn from a normal distribution.
with a certain probability
No, whether or not you reject, it doesn't give you P(normal) nor P(not normal)
I performed a KS test on my data and got a p < 2.2e-16, which seems to indicate my distribution is normal?
No, it tells you your data were not drawn from a normal distribution. (You didn't really need a test to tell you that, I could have told you that for free - and I wouldn't have even looked at your data.)
If I am to make predictions with this data, can I use normally distributed confidence intervals?
No goodness of fit test answers that question.
Unfortunately, neither can I right now, because you haven't give the sort of information required to judge; an answer would depend on a) the things you're doing, b) the properties you care about, and c) your tolerance for deviation from the nominal properties.
For example, it (a) if you're doing a t-interval for a mean, it's not very sensitive to moderate non-normality, but if you're doing an F-interval for a ratio of variances, its coverage properties are quite sensitive to non-normality; it depends on exactly what kind of interval you're talking about.