The fast of the one who has left the prayer is not correct and it is not accepted from him; because the one who abandons the prayer is a disbeliever, apostate; due to the Saying of Allaah the Most High:
But if they repent, perform the Prayer and give Zakaah, then they are your brothers in the Religion. (In this way) We explain the Ayaat in detail for a people who know. (At-Tawbah 9:11)
And the saying of the Prophet: 'The covenant between us and them (the hypocrites) is the prayer, so whomsoever abandons it then they have disbelieved'1
And the saying of the Prophet: 'Between and man and polytheism and disbelief is the abandonment of the prayer'2
And this was the general view of the Companions, if not their consensus.
Abdulllaah bin Shaqeeq rahimahullaah said - and he is from the well known Taabi'een -: 'The companions of the Prophet of Allaah sallAllaahu alayhi wa sallam never saw anything from amongst the actions, the abandonment of which is disbelief, except the prayer'.
And because of (all) this (evidence), if a person was to fast whilst he does not pray then his fast is rejected, not accepted and of no benefit for him with Allaah on the Day of Judgement. And we say to him, pray then fast, as for you fasting and not praying; then your fast is rejected from you, because the worship of the disbeliever is not accepted from him.
1: Reported by at-Tirmidhi, Kitaabul Imaan, Chapter: What is reported regarding abandonment of the Prayer, no 2621. And an-Nasaa'ee, Kitaabus-Salaat, Chapter: The Ruling upon abandoning the prayer, no 463. And Ahmad in "al-Musnad" (5/546). And al Haakim in "al-Mustadrak" (1/7) and he said "authentic chain of narration" and ath-Thahabee agreed with him. at-Tirmidhi said "Hasan Saheeh Ghareeb" 2: Reported by Muslim, Kitaabul Imaan, no (82)