Husband and Wife Sleeping In the Nude
Question

I would like to know whether sleeping naked with your spouse is permitted in Islam? If yes, then if you embrace each other while sleeping will you have to do ghusal before prayers or just wudoo would do?
Praise be to Allah.Is sleeping naked with the spouse permitted?
With regard to the first part of the question, it is permissible for the couple to sleep naked.  Allah says (interpretation of the meaning):
“And those who guard their chastity (i.e. private parts, from illegal sexual acts.) Except from their wives or (the slaves) that their right hands possess, for then, they are free from blame.” [al-Muminun 23:5-6]
Imam Ibn Hazm (may Allah have mercy on him) said: 
“Allah commands us to guard our chastity except from our wives or (the slaves) that our right hands possess; there is no blame in that case. This is general and includes seeing, touching and interacting.”  (al-Muhalla, 9/165) 
With regard to (evidence from) the Sunnah, it is narrated that ‘Aishah (may Allah be pleased with her) said: “The Messenger of Allah (peace and blessings of Allah be upon him) and I used to do ghusl from one vessel placed between me and him. He would hasten until I would say, ‘Leave some for me, leave some for me.’”(Narrated by al-Bukhari, 258; Muslim, 321 – this version narrated by Muslim). 
Al-Hafiz Ibn Hajar said:  
Al-Dawudi took this as evidence to indicate that it is permissible for a man to look at the ‘awrah of his wife and vice versa. This is supported by the report narrated by Ibn Hibban from Sulayman ibn Musa, who was asked about a man looking at his wife’s private parts. He said, I asked ‘Ata and he said: I asked ‘Aishah and she mentioned this hadith. Al-Hafiz said: this is the definitive report on this matter. 
There is another hadith from the Sunnah, in which the Messenger of Allah (peace and blessings of Allah be upon him) said: “Guard your ‘awrah, except from your wives or those (slaves) whom your right hands possess.” (Narrated by Abu Dawud, 4017; al-Tirmidhi, 2769 – he classed it as hasan; Ibn Majah, 1920. Narrated mu’allaqan by al-Bukhari, 1/508). 
Al-Hafiz Ibn Hajar said, commenting on this hadith: what may be understood from the phrase ‘except from your wives’ is that it is permissible for a wife to look at his (‘awrah) and by analogy that he is also permitted to look (at her ‘awrah). 
Ibn Hazm (may Allah have mercy on him) said: 
“It is halal for a man to look at his womenfolk’s private parts – his wife and his female slave with whom he is permitted to have intercourse – and it is permissible for them to look at his private parts. There is no reservation at all concerning that. The proof of that is the well known reports narrated from ‘Aishah, Umm Salamah and Maymunah, the Mothers of the Believers (may Allah be pleased with them), who narrated that they used to do ghusl with the Messenger of Allah (peace and blessings of Allah be upon him) to cleanse themselves from janabah (impurity following sexual activity) from one vessel. The report of Maymunah clearly indicates that he (peace and blessings of Allah be upon him) was uncovered, because in her report it states that “he (peace and blessings of Allah be upon him) dipped his hand into the vessel then poured water over his private parts and washed them with his left hand.” After this there is no need to follow the (mere) opinion of anyone. It is strange that some of those ignorant people who want to make things difficult allow intercourse but forbid looking at the private parts!” (al-Muhalla, 9/165) 
Shaykh al-Albani (may Allah have mercy on him) said: “Forbidding looking is a means of forbidding that which may lead to haram intercourse. If Allah has permitted a husband to have intercourse with his wife, how can it make sense that He would forbid him to look at her private parts?! No way.” (al-Silsilah al-Da’ifah, 1/353)
Does hugging the wife necessitate ghusl?
With regard to the ruling on taharah (purity) in this case, embracing one another whilst sleeping, so long as it does not lead to emission of maniy (semen) or intercourse, does not necessitate ghusl.
But if madhiy (prostatic fluid) is emitted, then the man has to wash his penis and testicles, and do wudu for prayer, and the woman has to wash her private part and do wudu likewise. 
And Allah knows best.
For more, please see these answers: 5560 , 21388 , and 45514