Have you tried searching Civil War Records on www.Fold3.com? I don't know if these two records are the person you are look for but a search on his name, refined by State of Massachusetts returned 68 records e.g., a service record for William McElroy, Reg. 46th, see URL: https://www.fold3.com/image/591292001 and a widows pension record filed by Emily F. Reed McElroy at URL: https://www.fold3.com/image/229181190?terms=William%20McElroy .
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Hello Mr. McElroy,
We were able to locate a Edward McElroy on ancestry.com under the Civil War pension index. However, to have access to the content, you will have to visit the National Archives in Washington D.C. to request the record in person. Or, you many order a reproduction of the pension file by mailing a NATF-85 form or submitting a request online.
William G McElroy - Reg 8 Indpt Batty MA L.A. - Appl # 998553; Widow App. 847112 Invalid Certificate 616703
Widow - Ellen A. McElroy
I hope this helps; good luck with your research.
Was there a Brother James McElroy as well, did find information of William McElroy he enlisted at 19 as a Pvt. On June 11th 1862 and mustered out in Nov 29th 1862. If you need anything else let me know.