I've found nothing to document Henry's birthyear as 1773, and the censuses indicate it must have been earlier.

Henry's parents are said to be this couple, but I've found nothing to document the connection:
Father:  Bartholomew "Barton" CARRICO
Mother:  Winifred "Kitty" PADGETT

1.  Marriage Record: 

2.  1790 Census Index/Images (online at Ancestry.com, Image #7 of 32 — list was roughly alphabetized; extracted by Diana Gale Matthiesen):¤•

1790

MD

Charles Co.

Roll M637_3

p. 13/554

Ln. 4

Henry Callico

2-1-2-0-1

These data indicate:

Gender and Type	Age Class	Therefore Born	Individuals Inferred
2 free white males	16 or over	in or bef. 1774	= Henry = ?
1 free white males	15 or under	1774-1790	= Son A
2 free white females	all ages	in or bef. 1790	= Susanna = Daughter A
1 slave	all ages	in or bef. 1790	= ?

The children here are said not to be Henry's based on Henry's supposed birthyear of 1773, but I have found nothing to document Henry's birthdate, so I see no reason not to assume these are his children.  (The male over age 16 is too old to be his son.)

3.  1800 Census Index/Images (online at Ancestry.com, Image #6 of 13; extracted by Diana Gale Matthiesen):¤•

1800

MD

Charles Co.

Trinity Par.

Roll 10

p. 515

Ln. 3

Henry Carricoe

40010-10010-00

These data indicate:

No. & Sex	Age Class	Therefore Born	Individuals Inferred
4 males	9 or under	1790-1800	= Cornelius (b. 1798) = John Stanislaus (b. 1794) = Son B
1 male	26-44	1755-1774	= Henry
1 female	9 or under	1790-1800	= Mary Harriet (b. 1790)
1 female	26-44	1755-1774	= Susanna

4.  Census Index:  U.S. Selected Counties, 1810 (Broderbund CD-313):  Henry CARRICO (or var.) not found, anywhere.

5.  1820 Census Index/Images (online at Ancestry.com, Image #38 of 52; extracted by Diana Gale Matthiesen):

1820

KY

Washington Co.

Roll M33_29

p. 72

Ln. 8

Susan Carrico

121200-02102-0-300

no slaves

These data indicate:

No. & Sex	Age Class	Therefore Born	Individuals Inferred
1 male	9 or under	1810-1820	= ?
2 males	10-15	1804-1810	= ? = ?
1 male	16-18	1801-1804	= Henry (b. ca. 1801)
1 male	19-25*	1794-1801	= Cornelius (b. 1797/8)
2 females	10-15	1804-1810	= Perdilia (b. 1807) = ?
1 female	16-25	1794-1804	= Harriet? (b. 1789/90)
2 females	45 or over	in or bef. 1775	= Susanna = ?
3	persons engaged in agriculture
*In the 1820 Census, the third column is age class 16-18 and the fourth column is age class 16-25, which means any individual in column three is duplicated in column four.  So, for an accurate representation of the household, the number in column three should be subtracted from the number in column four. If, after the subtraction, any individuals remain in column four, then they have to be age 19-25, thus creating a "new," more precise age class.

It looks like this may be our subject's widow, but that possibly another widow and her children are living with her.  Or are all the children Susan's?  Harriet, Cornelius, Henry, and Perdilia are not yet married, so should be in the household.

6.  Homer Edwin Carrico.  1981.  "The Carrico Family." Pp. 215-250 in Genealogies of Kentucky Families.  Genealogical Publ. Co., Baltimore, MD (Broderbund CD-185).  [This article is known to contain errors; consult with caution.]  Gives children Stanislaus, Delia, and Harriet, only.

7.  Leo Willett, ed. and compiler; Pearle (Willett) Curtsinger & Rachel Willett, eds.  1988.  A History of St. Jerome, Fancy Farm, Ketucky, 1836-1986.  St. Jerome Parish Community, Fancy Farm, KY (as cited by Linda Boorom, Archives of the CARRICO-DNA mailing list, online at RootsWeb.com):

p. 155- 156	Carrico Family (1) Cornelius Carrico s/o Henry & Susanna Carrico of Maryland...
p. 157	(Editor's Note: In a letter from William C. Morton, dated Apr 12, 1985, he wrote: "...by organizing and commenting on the known facts I believe that I have presented predominant and ample evidence that Mary Harriett (m. Stanislaus Toon in 1828), Stanislaus, John I. (m. Ann Montgomery in 1817), Cornelius (m. Teresa O'Bryan in 1823), Henry (m. Lucy Reed in 1831), and Perdelia (m. William Toon in 1829) were brothers and sisters to one another."  The parents of these six children were Henry & Susanna Carrico of Charles County, MD. by William C. Morton San Francisco, CA
I'm convinced that Stanislaus and John I. are the same individual (and it's John S., not John I.), which is why Morton found no wife for Stanislaus.  That they were the same man was very obvious from the census records, so you wonder about Morton's credibility.

8.  WorldConnect / Ancestry World Trees (online at RootsWeb.com/Ancestry.com).

9.  Messages posted to the Carrico Family Genealogy Forum (online at GenForum.com).