a good way to implement such a system would be to have a chairman of the UP. You could make it either a rotating chairmanship (like the EU presidency) or an elected chair (like the UN in Civ 4) or the chair could be the major race with the highest influence. The chair would then have the authority to table a resolution for a vote. In the interests of balance (not to mention variety) you could make it so once a resolution has been voted down it cannot be reintroduced for (say) 5 sessions. Also I think it would be a good idea to let the chairman put existing laws to the vote in order to have them repealed
To improve diplomacy you could set up the UP screen so that you could click on a race portrait to start diplomacy with them, but in addition to the standard tradeables there would be a "vote" section containing items like "Vote for A", "Vote for B" etc. That way you could buy alien votes.