Is Cross Validation needed for regression if you already know the predictors in your model?
You might.
(But probably not.)
The point of cross validation is to help you conduct model selection. You've already selected your model. One place where you might want to use cross validation is if you want to use some kind of regularization, but that might interfere with your ability to infer how each $X$ influences $Y$. (The regularized estimates are biased.)