No, they don't. If statistics are automatically created, it is because the optimizer can benefit from the single-column statistics of the cardinality estimates. It's simply an option. I'm guessing your real question may stem from statistics being automatically updated (as opposed to automatically create).
BOL definition for Statistics:
Statistics for query optimization are objects that contain statistical information about the distribution of values in one or more columns of a table or indexed view. The query optimizer uses these statistics to estimate the cardinality, or number of rows, in the query result.
There are many parameters for exactly WHEN statistics are marked as invalid (therefore updated when the next using query executes). Please see this blog post for a list of when this happens.
Take an example like this: You have a sequential table. You do 10 million sequential inserts at the end of the table. There is going to be zero fragmentation in that table. But the statistics will be forced as invalid (stale) and surely updated.
Therefore, the stats are updated. But there is no need to rebuild that clustered index (provided it's not a heap).
Often times, there will be inserts that aren't as sequential and this will cause page splits. So many inserts, much index fragmentation. Many inserts, stale statistics. So will stats be updated and the need for an index rebuild? Yes, absolutely.
But these two do not walk hand-in-hand.