He had a haraam relationship with a relative of his; is he obliged to marry her?

Firstly: 

Zina is a crime on its own, in terms of its sin and
consequences. None of its sin can be waived unless you repent sincerely to
Allah, may He be glorified and exalted, as He says (interpretation of the
meaning):

“And those who invoke not any other ilaah (god) along with
Allah, nor kill such person as Allah has forbidden, except for just cause,
nor commit illegal sexual intercourse __ and whoever does this shall receive
the punishment.

69. The torment will be doubled to him on the Day of
Resurrection, and he will abide therein in disgrace;

70. Except those who repent and believe (in Islamic
Monotheism), and do righteous deeds; for those, Allah will change their sins
into good deeds, and Allah is Oft‑Forgiving, Most Merciful

71. And whosoever repents and does righteous good deeds; then
verily, he repents towards Allah with true repentance”

[al-Furqaan 25:68-70]. 

Although marriage may offer concealment for what occurred
between you of immoral and indecent actions, it is not a prescribed remedy
for the sin and shame of zina. The remedy for that is sincere repentance and
mending one’s ways as much as possible in the future, and doing a lot of
good deeds, in the hope that Allah may bless you by accepting your
repentance and good deeds. 

See also the answers to questions no.
147808 and 14381 

Secondly: 

In that case, we say to you: 

If what you mean when you say that this girl is not religious
is that she is promiscuous and morally corrupt, and that she does not
respect the religion of Allah or protect herself (from shameful deeds) and
is not chaste, then forget about this girl and her ways; focus on yourself,
repent properly to your Lord, adhere to His commands, do as many good deeds
as you can, and keep yourself chaste by marrying a woman who is good and
righteous. 

But if this girl has repented from what happened between you,
and it was the Shaytaan who tempted her to do that, as he tempted you,
because she was infatuated with you, or because of a mistake on her part,
and it seems to you from her attitude that she regrets having disobeyed her
Lord, and the worst that can be said is that she is careless with regard to
hijab, or with regard to some matters, then there is nothing wrong with you
marrying her, after you and she have repented. In fact we encourage you to
do that, because you and she have in common the crime of zina, and perhaps
it was you who tempted her or promised her marriage, or you led her on,
until you fell into that sin. In that case it is neither fair nor decent to
leave her to bear alone the shame and consequences of what you did together,
and let her carry that burden on her own, whilst you carry on with your life
as if you never did anything! 

Undoubtedly your marrying her would come under the heading of
complete concealment that Allah loves for His slaves, and striving to
relieve the distress of one who is distressed, so long as he has repented
from his sin. 

If you think that your family will object, then try to
convince them that marriage to relatives is not all bad, and make her show
them that which will endear her to your family. Encourage her to repent
sincerely and to show the effects of her repentance in her outward
appearance and her conduct. 

If it is not possible for you to marry her, and your families
object to such a marriage, and you think that this marriage may be a cause
of severing ties of kinship or disobedience towards parents, then in that
case there is no blame on you if you end your relationship with her and
focus on your own affairs, and leave her to her own affairs, after making a
concerted and sincere effort to conceal her sin, keep her chaste and protect
her. And Allah will take care of her affairs, rectify her condition, enable
her to repent sincerely and bless her with a husband who will keep her
chaste. 

And Allah knows best.