You could alternatively try defining Gaussian measures as [$2$-stable][1] distributions. This does remove any reliance on finite dimensional projections, and even removes reference to topology. Let $V$ be a measurable vector space (by which, I mean a real vector space $V$ with sigma-algebra $\mathcal{F}$ with respect to which addition and multiplication are measurable).

A probability measure $\mu$ on $V$ is then a *centered* Gaussian iff, for any independent pair $X,Y$ of $V$-valued random variables each with measure $\mu$, then $aX+bY$ also has measure $\mu$ for all $a,b\in\mathbb{R}$ with $a^2+b^2=1$.

If $A$ is a (measurable) affine space with underlying vector space $V$, then we could similarly say that $\mu$ is Gaussian iff there exists an $m\in A$ such that $X-m$ is a centered Gaussian on $V$ for a random variable $X$ with measure $\mu$.

Several facts should then follow quickly from this:

 - Affine maps take Gaussians to Gaussians, and linear maps take centered Gaussians to centered Gaussians.
 - Linear combinations of independent (centered) Gaussians are again (centered) Gaussians.
 - On separable Banach spaces, the definition is equivalent to the standard one as measures whose one-dimensional projections are Gaussian. More generally, this holds for any locally convex space on which addition is jointly Borel measurable (e.g., separable [Fréchet spaces][2]).
 - The definition even makes sense for, e.g., separable [F-spaces][3] which can have trivial dual. (Whether it is actually useful to consider Gaussians in such spaces is another question).

This seems to give an answer first paragraph of the question, and does not depend on projections. I'm not sure if it is going in the direction that the question was asking for though, as it says nothing about the stronger form of the question further down and didn't mention covariance operators at all.

  [1]: https://en.wikipedia.org/wiki/Stable_distribution
  [2]: http://en.wikipedia.org/wiki/Fr%C3%A9chet_space
  [3]: https://en.wikipedia.org/wiki/F-space