The connections between the Washington and the Lewis families span generations. The Washingtons and Lewises were related by marriage and by birth. Both families descend from Augustine Warner II (1642-1681), Speaker of the Virginia House of Burgesses. The chart below is a greatly abbreviated version of the family tree that shows the direct connection between the two families.
Mildred Warner and Elizabeth Warner were sisters. Their father, Augustine Warner II, was a common great grandfather to Fielding Lewis and Catharine, Betty, and George Washington.
John Washington (Catharine's father) was a first cousin to John Lewis (Fielding's father).
Augustine Washington (Betty's father) was also a first cousin to John Lewis (Fielding's father).
Catharine and Betty were first cousins. Catharine and Fielding were second cousins. Betty and Fielding were also second cousins. (Although not shown on the chart, since George and Betty Washington were brother and sister, George and Fielding were also second cousins as well as being brothers-in-law.)
In 1799, Lawrence Lewis, ninth child of Fielding and Betty, reconnected the two families when he married Eleanor (Nelly) Parke Custis. She was the granddaughter of his Uncle George Washington's wife, Martha Dandridge Custis Washington, by her first marriage to Daniel Parke Custis.