Check if contiguous date interval exists

First, we combine intervals that overlap to find all the contiguous "islands" of the intervals:

with c as
  ( select *, max(end_date) over (order by start_date
                                  rows between unbounded preceding
                                           and 1 preceding)
                as previous_max
    from my_table
  )
select start_date, 
       coalesce(lead(previous_max) over (order by start_date),
                (select max(end_date) from my_table)
               ) as end_date
from c 
where previous_max < start_date 
   or previous_max is null ;

After that, it's easy to check if a given interval in completely surrounded by one of the found contiguous islands.

   with c as
  ( select *, max(end_date) over (order by start_date
                                  rows between unbounded preceding
                                           and 1 preceding)
                as previous_max
    from my_table
  ) ,
cont as
  ( select start_date, 
           coalesce(lead(previous_max) over (order by start_date),
                    (select max(end_date) from my_table)
                   ) as end_date
    from c 
    where previous_max < start_date 
       or previous_max is null  
   )
select *
from cont
where tsrange(start_date, end_date)
       @>                             -- contains
      tsrange('2015-04-02 22:10:00', '2015-04-02 22:30:00') 
limit 1 ;

Test at SQLfiddle

This query returns all gaps in the data that overlap with the given time range. Also covers leading, trailing or total (covering) gaps:

WITH input(t1, t2) AS (SELECT '2015-04-02 22:00'::timestamp
                            , '2015-04-02 23:00'::timestamp)  -- t1 < t2
,    gap  AS (
   SELECT max(end_date) OVER (ORDER BY start_date
             ROWS BETWEEN UNBOUNDED PRECEDING AND 1 PRECEDING) AS start_gap
        , start_date AS end_gap
        , end_date  -- for possible trailing gap
   FROM   my_table
   WHERE  end_date   >= (SELECT t1 FROM input)  -- exclude rows early
   AND    start_date <= (SELECT t2 FROM input)
   )
SELECT start_gap, end_gap
FROM   gap
WHERE  end_gap > start_gap  -- gap within time range
   OR  start_gap IS NULL AND end_gap > (SELECT t1 FROM input) -- leading gap

UNION ALL
SELECT max(end_date), NULL  -- trailing gap
FROM   gap
HAVING max(end_date) < (SELECT t2 FROM input)

UNION ALL
SELECT NULL, NULL  -- total gap
WHERE  NOT EXISTS (SELECT 1 FROM gap)
ORDER  BY end_gap;

If nothing is returned, the check succeeds:

checks if a contiguous date interval between two given dates exists?

Only needs a single scan over the base table and a single window function. Should be fast.

If you are not familiar with window functions, read this chapter of the manual.

Related:

• Compare multiple date ranges

About OVERLAPS:

• Find overlapping date ranges in PostgreSQL

Function counting gaps

For a quick answer to your question without details, the query can be simplified. Also parameter handling is more convenient in a function:

CREATE OR REPLACE FUNCTION f_find_gaps(t1 timestamp, t2 timestamp)
  RETURNS int AS
$func$
WITH gap AS (
   SELECT max(end_date) OVER (ORDER BY start_date
             ROWS BETWEEN UNBOUNDED PRECEDING AND 1 PRECEDING) AS start_gap
        , start_date AS end_gap
        , end_date  -- for possible trailing gap & total gap
   FROM   my_table
   WHERE  end_date   >= t1  -- exclude rows early
   AND    start_date <= t2
   )
SELECT count(*)::int
     + CASE WHEN (SELECT max(end_date) >= t2 FROM gap) THEN 0 ELSE 1 END
     -- covers trailing gap & total gap
FROM   gap
WHERE  end_gap > start_gap  -- gap within time range
   OR  start_gap IS NULL AND end_gap > t1  -- leading gap
$func$ LANGUAGE sql;

Call:

SELECT f_find_gaps('2015-04-02 22:00', '2015-04-02 23:00');

Returns the number of gaps. 0 means, your range is covered.