When you have known means / variances, this classifier amounts to just finding the likelihood of your sample under the two models and choosing the one that's greater. I don't use R, but it looks like
dmvnorm will find likelihoods for you;
dnorm2d is bivariate-specific but you have to subtract the mean off and maybe divide by sigmas yourself, while
dmvnorm will do those for you but you have to make the 2x2 covariance matrix yourself.
I'm not sure what you mean by the variables being independent: that you're dealing with IID samples of pairs, or that the two elements of the vector are independent? In the latter case, you could also just use 1D normal likelihoods and multiply them.