Did you ask this question on physics forums? If so I gave an answer there.
No it has to be if P(Y|X) is proportional to P(X|Y)*P(Y) and if P(X|Y) and P(Y) are both Normal then P(X|Y) also has to be normal.
For a more indepth explanation look up Bayesian statistics with Normal Likelihood and a Normal conjugate prior to see the details.
If you want to prove it, take two PDFs that are Normally distributed f(X) and g(X) and then show that f(X)*g(X) also has a normal distribution. This is a very simple exercise and would do you well to understand what is going on.
Someone else tells me
Look at the 2-d density function for dependent normal variables. You will see that, except for singular cases, that the form of the integrand is symmetric in x and y (except for constants). Therefore if one of the conditional distributions is normal, the other must be.
Singular case - correlation = 1.
Is he wrong then?