The problem isn't vertical space. If vertical space was added below the picture or above the proof, they would be on different lines. bussproofs does add vertical space above and below proofs and this can be adjusted, but it is not applicable to proofs displayed inline.
The issue is rather that bussproofs centres the proof vertically with respect to the current baseline. You can adjust the vertical positioning of the tikzpicture and/or the proof as desired, but adjusting the relative positioning of the proof is probably simplest in this case as it would be rather tricky to adjust the alignment of the picture relative to the baseline of the proof's conclusion.
bussproofs provides vertical alignment options for inline proofs: \normalAlignProof, \centerAlignProof and \bottomAlignProof. Somewhat confusingly, it is the first which centres the proof vertically. The second sets it lower and the third aligns the baseline of the conclusion with the current baseline. So that's the option we want to use here.
\documentclass[border=5pt]{standalone} \usepackage{tikz} \usepackage{bussproofs} \begin{document} \begin{tikzpicture} \draw (0,0) circle (0.15em); \end{tikzpicture} \bottomAlignProof \AxiomC{} \UnaryInfC{$\Gamma$} \DisplayProof \end{document}
Note that if you didn't know this or bussproofs provided no such option, you could put the proof into a box without a frame, since the desired alignment is a side effect of the frame rather than a direct result. It works because, in order to frame the proof, LaTeX puts it in a box aligned with the current baseline. A frameless box, therefore, would work just as well as a framed one. As always, however, finding just the right box construct to do what you want without doing half a page's worth of calculations involving heights, depths, baselines and printers' measures is not entirely straightforward, so it is fortunate bussproofs's author has saved us the trouble.
Incidentally, should you find bussproofs insufficient for your needs, note that there are newer packages available. ebproofs uses a similar syntax, for example, but offers a bit more flexibility. If you don't know about LaTeX for Logicians, you should ;). Coverage of natural deduction is on this page.
