Bogachev's book on Gaussian measures states: "A measure $\gamma$ on a locally convex space $X$ is Gaussian iff its Fourier transform has the for $$\hat{\gamma}(f) = \exp \bigg{(}iL(f)-\frac{1}{2}B(f,f)\bigg{)}$$ where $L$ is a linear function on $X^{*}$ and $B$ is a bilinear function on $X^{*}$ such that the quadratic form $f \mapsto B(f,f)$ is nonegative.
Now, my question is concerned about the opposite direction. Suppose I have defined a bilinear map $B:\mathcal{S}'(\mathbb{R}^{d})\times \mathcal{S}'(\mathbb{R}^{d})$. Do we have any representation theorem for $\exp\bigg{(}-\frac{1}{2}B(f,f)\bigg{)}$? Can we write it as a Fourier transform of some measure on $\mathcal{S}(\mathbb{R}^{d})$ so that, because of Bogachev's statement, this measure is Gaussian?
The answer is yes but you are forgetting a crucial hypothesis: the bilinear form $B$ must be a continuous map from $\mathcal{S}'\times\mathcal{S}'$ to $\mathbb{R}$. For a proof of the result (analogue of Bochner-Minlos for probability measures on $\mathcal{S}$ instead of $\mathcal{S}'$) see Theorem 4.3 in the article "Generalized random fields and Lévy's continuity theorem on the space of tempered distributions" by Biermé, Durieu and Wang.