No, is the short answer to your question providing if one has performed ghusl correctly, and washed each part and completed the 3 faraidh, then they do not have to do wudhu. Below is the faraid of the Ghusl

Ghusl is performed in a manner which is very specific:
Gargling – so that each part of the mouth is not left unwashed
Placing water in the nostrills – making sure water reaches the soft part of the bone
Pouring water all over the body – making sure that not even a hair is left unwashed, carefully pouring water and rubbing to make sure water reaches everywhere.

Method Of Bathing
The method of bathing is as follows:
1. Perform the intention of bathing (Ghusl) then first of all wash the hands upto the wrists three times.
2. Then wash the private parts, whether there is any impurity stuck on them or not.
3. Then wash any part of the body which has impurity stuck on it.
4. Then perform Wuzu as performed in Namaz but do not wash the feet. However, if you are sat on a stool or a stone etc. then wash the feet also.
5. Then rub water into the body as you would rub oil.
6. Then wash the right hand shoulder three times.
7. Then wash the left hand shoulder three times.
8. Then wash the head and all of the body three times.
9. Then move away from the place of bathing and if you did not wash the feet when performing Wuzu, wash them now. Do not face the Qibla when bathing. Ensure that you rub your hands with water all over the body whilst scrubbing. Do not bathe where someone can see you. If this is not possible then it is necessary for you to cover the body from the navel to the knees (for men). If this is not possible then perform Tayammum (see later text). Do not talk when bathing or pray any prayers. After bathing, to dry the body with a towel is perfectly allowed.

Ø To bath naked in a careful hidden place (where no-one can see) is allowed. It is very important that women are very careful of this. After bathing wear clothes as soon as possible. Whatever aspects are Sunnat and Mustahhab in Wuzu are also the same in bathing except, if you are bathing naked then do not face Qibla and if you are wearing a cloth then there is no problem. There are three obligatory aspects in the method of bathing that has been shown, without which the bathing will not count and one will remain unclean and the rest are either Sunnat or Mustahhab and should not be left out, but if they are left out the bath will still count.

Three obligatory aspects of bathing :
There are three obligatory aspects in bathing, and are as follows;
1. To perform mouthwash so that every single part of the inside of the mouth from the beginning of the lips upto the throat is cleaned with water. Water must reach the gaps between the teeth, the gums, all the sides of the tongue and the edge of the throat. If you are not fasting then you must gargle so that the water reaches clearly all parts. If there is anything stuck in the teeth (like strands of meat, the skin of a betel nut, the leaf of a paan etc.) then unless it is impossible to release or would cause serious pain it is necessary that they are removed, because without doing this the bath won`t count and therefore the Namaz won`t count.
2. To clean the nose out with water. Meaning to suck up water into both nostrils until it reaches the bone, so that not even a hair or it`s equivalent size remains dry, otherwise the bath will not count. If the nose is pierced then the water must reach the hole as this is also necessary. If mucus has dried in the nose then to release it is necessary and to wash the nostril hair is also necessary.
3. To wash every single part of the body. Meaning to make sure water washes all the body upto and including the soles of the feet, ensuring that every hair and every pimple is washed, because even if only one hair or it`s point`s equivalent remains dry, the bath will not count. Notice. A lot of people bathe by wearing an unclean cloth and think by bathing they will become clean and at the same time wash the cloth that they are wearing. This is not the case because when they rub their hands on it they actually spread the impurity all over and therefore make the whole body, cloth and the container of water unclean. This is why before bathing it is important to wash the impurity stuck on the body or on the clothing, otherwise they won`t clean the body but in fact make everything that they touch unclean. This is possible if they are bathing in a river or sea and the impurity is such that it will flow away without the need for rubbing or scrubbing, if this is not so then it will remain a problem.

What aspects makes it obligatory to bathe :
Ø There are five aspects that make it obligatory for a person to have a bath, and they are as follows;
1. The ejaculation of sperm from it`s place with lust.
2. To have a wet dream. Meaning nocturnal emission that is the releasing of sperm whilst sleeping.
3. The head of the penis to enter the vagina, whether it be with or without lust, with or without orgasm, the bathing would be obligatory on both.
4. To become clean from menstrual pause (period).
5. To become clean from blood after child birth.
Ø If sperm was ejaculated from it`s place but not due to lust, i.e. it was done by lifting a heavy load or falling from a height, then to bathe is not obligatory, however, the Wuzu is broken.
Ø If semen dropped but was very thin and it came out whilst urinating or on it`s own accord without lust then the bath does not become necessary but the Wuzu will break.
Ø To bathe on Fridays, Eid days, on the day of Arafah (9th Zil Hajj) or when wearing the Ehraam is Sunnat.

Things that can and cannot be done when bathing is obligatory :
Ø It is strictly forbidden to enter a mosque, to perform Tawaf (of Kaa`ba), to touch the Holy Quran, or any part of the Quran whether it is in sections or volumes [Hadiya, Alamgiri] , to pray the Quran whilst viewing it without touching it, to pray the Quran verbally, to wear or touch a ring with a word of the Quran written on it when it is obligatory for you to bathe.

If the Holy Quran is wrapped in it`s cloth (Juzdaan) or has a napkin wrapped around it, then there is no problem touching it without bathing [Hadiya, Hindiya]. If a verse of the Quran is prayed but the intention was not to pray as part of the Holy Quran, then there is problem. For example, to say `Bismilla Hirrahma Nirrahim` for auspisciousness, or to say `Alhamdo Lillahe Rabbil Aalameen` for thankfulness or to pray when in trouble or worried `Inna Lillahe Wa Inna Ilahi Raji`oon` or for praise the Surah Fatiha was prayed or Aayatul Qursi was prayed or another verse was prayed, however, the intention was not to pray a verse of the Quran, then there is no problem [Hindiya etc.].
Ø To touch the Holy Quran or any verse of the Quran without ablution (but not in need to bathe) is forbidden, however to pray it verbally is no problem.
Ø To see the Holy Quran , whether the words are seen and then prayed in the heart is allowed in any situation.
Ø To touch the Fiqah, Hadith or Tafsir books is Makruh. Which water is allowed to bathe or perform ablution and which is not Rainwater, sea water, ocean water, river water, stream water, canal water, well water, large pool or large lake or flowing water, snow and hailstone water is all allowed to be used to bathe or perform ablution or to clean impurities.
Ø If an impurity falls in a large pool that cannot be seen, e.g. alcohol, urine, etc. then you can perform ablution from wherever you want. If an impurity falls in that can be seen, e.g. stools, dead animal etc. then it is better to perform ablution away from where the impurity is.
Ø Many people can perform ablution together in a large pool, even though the used water is falling back in. However, mucus, phlegm, spittle should not be added back in the water as it is against purity.
[Answer provided by: Mawlana Mohammed Kalim]