Yes, if a state kept Obama off the ballot because of constitutional eligibility it would definitely allow Obama to take the issue to court, although if only one or two GOP states do it Obama may choose to ignore it (even if he was actually born in Hawaii) so as not to risk the court declaring him ineligible.
As for a major-party presidential candidate not being on the ballot in a state, it has happened more often than you think. Incumbent President Lyndon Johnson was not on the ballot in Alabama in 1964 (Goldwater carried the state, and “unpledged Democrat electors” finished second), and incumbent President Truman was not on the AL ballot in 1948 (Strom Thurmond was the nominee of the AL Democrat Party and carried the state). In 1912, incumbent President Taft was not on the ballot in SD, while former President Theodore Roosevelt, nominee of the Progressive (”Bull Moose”) Party and second-place finisher that year was not on the ballot in OK (the only two presidential candidates on the ballot in all 48 states that year, and thus the first in history to be on 48 state ballots, were Democrat Woodrow Wilson and Socialist Eugene Debs). And in 1892, former President Grover Cleveland gained back the White House despite not being on the ballot in a few states out West in which Populist Party candidate James Weaver was stronger than Cleveland and was the “fusion candidate” against Republican President Benjamin Harrison. Off the top of my head I can’t think of any other post-1860 major-party nominee to be denied ballot access, but maybe there were one or two others as well.
Argh, LBJ in AL in ‘64. I don’t know why I momentarily forgot that... Electors there refused to vote for JFK in ‘60, too, casting them for Harry Byrd and Thurmond (as did MS).
Not being on the ballot in 10 GOP leaning states won’t hurt that much. Where it could possibly hurt the most would be during the primary. If a strong Dem challenger is on the ballot in ten states that O isn’t it could swing the nomination.