What is the difference between referential integrity and the indexes
that support it?

REFERENTIAL INTEGRITY is a database concept that ensures that
relationships between tables remain consistent. When one table has a
foreign key to another table, the concept of referential integrity
states that you may not add a record to the table that contains the
foreign key unless there is a corresponding record in the linked
table. It also includes the techniques known as cascading update and
cascading delete, which ensure that changes made to the linked table
are reflected in the primary table.
An INDEX is a database feature used for locating data quickly within a
table. Indexes are defined by selecting a set of commonly searched
attribute(s) on a table and using the appropriate platform-specific
mechanism to create an index
As you can see there is a great divide. First is just a concept and
the second one is a feature that accomplish to locate data as quick as
possible. They don't have anything in common.

It's possible to add a constraint on a column (which may be an index
column) in order to maintain referential integrity, if the column is a
foreign key: and keys are usually (but not always) indexed for fast
lookup, because they will have the value that uniquely identifies a
row in the table.. is this what you were thinking of? E.g (SQL 3)

CREATE TABLE department
(department_number CHAR(4) CONSTRAINT prim_dept PRIMARY_KEY,
 department_name VARCHAR(10));

ALTER TABLE employee
ADD (CONSTRAINT emp_dept_fkey FOREIGN KEY(department_number)
REFERENCES department(department_number));