Did George Washington name the state of Washington?

No. At the time George Washington died in 1799 the US only had 16 states. The rest of the continent was held by France, Spain, and Britain.

What's now Washington state was part of a much larger region called the Oregon Territory. The OT was jointly administered by the US and Britain from 1818 to 1848, That year the territory was formally split into separate US and British Canadian sections. A few years after that the US section of was further split; the southern part kept the name Oregon Territory and the northern part became the Washington Territory in honor of George Washington.

The Oregon and Washington Territories were subdivided yet again during the years surrounding the US Civil War, forming the current states of Washington, Oregon, and Idaho as well as adding to the area of Montana and Wyoming.