As Thorsen cautioned in response #12, ONLY do this with a clear conscience. Depending on how one organizes data on an external hard drive, finding data more quickly may be more important than turning off PC to remove external hard drive.
As Melissa has already done, right-click on the external hard drive and left-click "properties". The attached screenshot is what one should get. Near the bottom with the last tick-box is what will indicate whether the external drive is indexed or not. If it is ticked, it is indexed. In my screenshot, it's unticked and therefore not indexed.
Depending on the amount of data already indexed, it may take an hour or MORE to "un-index" the drive and prevent this drive from being indexed in the future. Set aside time for it.
For argument's sake, if indexing is not the issue and ejecting safely is still a problem after un-indexing the external hard drive for you, Melissa, you can return to "properties", tick the box and return to normal indexing of this external hard drive.