Reciting a soorah after al-Faatihah is not obligatory

1 – Reciting Qur’aan after al-Faatihah in prayer is not
obligatory, either in a fard (obligatory) prayer or a naafil
(supererogatory) prayer, whether the prayer is one in which Qur’aan is
recited out loud or one in which it is recited silently, whether one is
joining the prayer late or otherwise. 

It was narrated that ‘Ata’ said: Abu Hurayrah said: In every
prayer there is recitation.  What we heard the Prophet (peace and
blessings of Allaah be upon him) say, we tell you, and what he did not tell
us about we cannot tell you. Whoever recites Umm al-Kitaab (i.e., Soorat
al-Faatihah) has done enough, and whoever does more than that, that is
better. 

Narrated by al-Bukhaari, 738; Muslim, 396. 

Al-Nawawi said: 

The phrase, “Whoever recites Umm al-Kitaab (i.e., Soorat
al-Faatihah) has done enough, and whoever does more than that, that is
better” indicates that reciting al-Faatihah is obligatory, and that nothing
else will do.  

It also indicates that reciting a soorah after al-Faatihah is
mustahabb. There is scholarly consensus on this point with regard to Fajr,
Jumu’ah and the first two rak’ahs of every prayer, and this is Sunnah
according to all scholars. Al-Qaadi ‘Iyaad (may Allaah have mercy on him)
narrated that some of the companions of Maalik said that reciting a second
soorah is obligatory, but this is an odd view which is to be rejected. 

With regard to reciting an additional soorah in the third and
fourth rak’ahs, there was a difference of opinion among the scholars as to
whether it is mustahabb or not. Maalik (may Allaah have mercy on him)
regarded it as makrooh. Al-Shaafa’i (may Allaah be pleased with him)
regarded it as mustahabb in his new opinion but not in his old opinion; the
old opinion in this case is more correct. 

Sharh Muslim, 4/105, 106. 

Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said: 

According to the majority
of scholars, reciting another soorah (after al-Faatihah) is Sunnah and is
not obligatory, because it is obligatory to recite only al-Faatihah. 

Al-Sharh al-Mumti’, 3/103 

If you stand up to complete your prayer after the imam has
said the tasleem, then the correct scholarly view is that the rak’ahs in
which you caught up with the imam form the first part of your prayer. See
question no. 23426.
If what remains of your prayer is the third and fourth rak’ahs, then you
should recite al-Faatihah only; if what remains is the second and subsequent
rak’ahs, then in the second you should recite al-Faatihah plus another
soorah, and in the remaining rak’ahs you should recite al-Faatihah only. 

It is permissible for a worshipper to recite another soorah
after al-Faatihah in the third and fourth rak’ahs, but he should do that
only sometimes; it was narrated that the Prophet (peace and blessings
of Allaah be upon him) did this according to a hadeeth narrated by Muslim
(452). 

And Allaah knows best.