An example of a family tree showing Adam to Elizabeth II http://www.originofnations.org/Royals/royal_line.html
If King David has any descendants alive today, then Queen Elizabeth II would indeed be his descendant... but so would every other human being on Earth. This is due to pedigree collapse.
However, that specific genealogy looks highly suspect to me because most of the kings of Ireland and Scotland on that family tree are legendary at best.