Source: his 1920 Passport Application on Ancestry.com, where he gives his DoB as 1873, his birth in Lislea, his residence in New York, his job as a stableman and his reason for going back to Ireland as to "settle my estate". As it happens, there is a Robert Gordon, son of Robert Gordon Jnr, of Lislea, whose father did indeed die in 1920. However, he was probably born circa 1860 rather than 1873 as he is likely to be the first son of Robert Jnr, whose father was Robert Snr, although we do not actually have any record of his birth.
Robert's DoB is confirmed by the 1920 census, where he is a widower and in the 1900 US census, he is given as married to a Kate, with a son called Thomas, aged 19.
In 1920 census he says he immigrated in 1889. In 1910 census, he says he immigrated in 1891.
VIEW 1920 United States Federal Census
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Name: Robert Gordon
[Robert Garden]
Age: 47
Birth Year: abt 1873
[abt 1875]
Birthplace: Ireland
Home in 1920: Manhattan Assembly District 5, New York, New York
Street: 50th Street
Residence Date: 1920
Race: White
Gender: Male
Immigration Year: 1889
Relation to Head of House: Head
Marital Status: Widowed
Father's Birthplace: Ireland
Mother's Birthplace: Ireland
Native Tongue: Irish
Able to Speak English: Yes
Occupation: Foreman
Industry: Stable
Home Owned or Rented: Rent
Naturalization Status: Papers Submitted
Able to Read: Yes
Able to Write: Yes
Neighbors: View others on page
Household Members:
Name Age
Robert Gordon 47
Thomas Gordon 19
Name: Robert Gordon
Age in 1910: 34
Birth Year: abt 1876
Birthplace: Ireland
Home in 1910: Manhattan Ward 19, New York, New York
Street: East 61st Street
House Number: 254
Race: White
Gender: Male
Immigration Year: 1891
Relation to Head of House: Head
Marital Status: Married
Spouse's Name: Kate Gordon
Father's Birthplace: Ireland
Mother's Birthplace: Ireland
Native Tongue: English
Occupation: Foreman
Industry: Livery Stable
Employer, Employee or Other: Wage Earner
Home Owned or Rented: Rent
Farm or House: House
Naturalization Status: Papers
Able to Read: Yes
Able to Write: Yes
Years Married: 16
Out of Work: N
Number of weeks out of work: 0
Neighbors: View others on page
Household Members:
Name Age
Robert Gordon 34
Kate Gordon 34
Mary C Gordon 16
Thomas Gordon 9
John Walsh 28
Thomas Connetly 25
James Davy 24
Name: Robert Gordon
Age: 27
Birth Date: Aug 1872
Birthplace: Ireland
Home in 1900: Manhattan, New York, New York
Ward of City: 22nd
Street: West 53rd Street
House Number: 415
Sheet Number: 5A
Number of Dwelling in Order of Visitation: 7
Family Number: 103
Race: White
Gender: Male
Immigration Year: 1891
Relation to Head of House: Head
Marital Status: Married
Spouse's Name: Kate Gordon
Marriage Year: 1894
Years Married: 6
Father's Birthplace: Ireland
Mother's Birthplace: Ireland
Years in US: 9
Occupation: Stableman
Months not employed: 0
Can Read: Yes
Can Write: Yes
Can Speak English: Yes
House Owned or Rented: R
Farm or House: H
Neighbors: View others on page
Household Members:
Name Age
Robert Gordon 27
Kate Gordon 27
Mary C Gordon 4
Hannah Walsh 26