Is Gaussian noise equal to white noise?

No, they are completely orthogonal concepts. The probability distribution says nothing about the frequency content, and the power distribution across frequency says nothing about the sample probability distribution. You have to specify both.

As Dave (and Brian) said: two totally different concepts. One doesn't imply the other. This is homework, and you should research it well! Getting the difference between (auto)correlation/PSD and amplitude distribution straight is a critical thing. If this isn't clear to, you should probably ask your professor/teacher (if you have one) – it's easier to explain if one has a didactic "framework" to work with.

There's one thing that's special about Gaussian noise w.r.t. to correlation, and that if random variables (for example, noise measurements from different times) are jointly uncorrelated (and that's a big restriction!), then they are independent.

For all other distributions, lack of correlation does not imply independence.

This is a property about circularly symmetric gaussian (\$\sim\mathcal{CN}\$) noise that allows us to do a lot of mathematical transforms on it (e.g. correcting the phase of a received signal) and still have independent noise components, and that is what's necessary for a lot of estimators to actually work optimally. So, hurray for circularly symmetric gaussian noise!

Gaussian noise definitely does not imply white noise, because Gaussian noise can have an arbitrary (not necessarily flat) frequency spectrum.

However, contrary to the other answers, there is a sense in which white noise implies Gaussian noise, if the noise is white to arbitrarily high frequencies (arbitrarily small time scales). Or more practically, if our measurements average the noise over time intervals much longer than its correlation time. In this case, the central limit theorem says that the measured noise amplitude, being composed of many independent contributions (with finite variance for physical reasons), converges to a Gaussian distribution.

EDIT: How much longer than the correlation time is needed for the central limit theorem to converge depends on the statistics of the noise. John Doty's comment points out that it does not happen quickly for white noise consisting of pulses that follow a Poisson process. In this case, the amplitude has a highly skewed distribution that is mostly concentrated on zero. This is a "worst case" for the central limit theorem. Averaging over a few pulse widths (correlation times) doesn't make it Gaussian; we have to average over longer than the mean interval between pulses. When we do this, we start to get a less skewed Poisson distribution that is approximately Gaussian. So it still holds that if measurements are averaged over long enough times, white noise looks Gaussian.