Take Things "Off Line"
There are times when you can't resolve a situation in a meeting: this is particularly the case where problems involve sensitive personal issues, which shouldn't be discussed "in public".
In this case, you'll need to acknowledge the disagreement, and arrange a specific meeting to address the issue later on.
Finally, remember that sometimes it simply isn't possible for everyone to be happy with an outcome. If you've given everyone a fair chance to express their opinions, and you've gone through a fair decision making process (where appropriate), don't take it personally when people are unhappy. This also applies to you – if you're the one who's unhappy with a decision.