I think he may have been born earlier than 1770 unless his father was named Robert also as listed
LINK in CLONMEL DIRECTORIES 1788-1930 as Dillon Street in 1788
Henry entered TCD aged 16 as in this
LINK on page 148; implying born c 1820
In 1829 Robert is described as a magistrate
Henry's father is given in the last link as Robert Constable of Prior Park, Clonmel
His will is referenced on line - he died in 1830 -
LINKThere is access to the will on a paid site - difficult to read old writing but mentions a son William, Henry and daughter Jane; also a sister Catherine Wilkinson but no mention of a son Robert or wife however Tipperary Clans have an entry in 1825 for Robert Constable Jun.; eldest son of Robert MD; place Clonmel which implies at least one brother
There is also under UK, British Army and Navy Birth, Marriage and Death Records, 1730-1960 for Robert Constable; WO 42: Officers Birth Certificates, Wills and Personal Papers Piece 009: C 402-511 (1776-1881) reference to Robert's widow Anna of Prior Park, Clonmel, he shown as an Apothecary to the Forces and that they married on Jul 24 1811 in Rathconon? It goes on to state she was Anne Carey daughter of Rev. Richard Carey of Rathronan (is more likely this time) It also states that Dr. Constable was in his 61st year (d. 1830 => bc 1769)
Unless this is a second marriage it rules out the Ann Lanphier 1806 marriage
Linking these records may give you what you are looking for
Also if you locate the marriage of Catherine Constable to unnamed Wilkinson you may locate their parents
