Such a child is regarded after his mother, as elborated in the Gemara (Yevamos pages 16-17, and Kidushin page 68) "Your son (meaning grandson) from a Yisraelit is called your son (meaning that he is called Yisrael or "Jewish" today), and your son from the gentile woman (meaning she who had a child with your son) is not called your son (meaning he is not considered Yisrael)"
And the Shulchan Aruch (4, 5) also states that he does not require a Giur ("הנולד מישראלית וגוי נחשב כישראל לכל דבר ואינו צריך גרות")
And so from the words of the Rambam (הל' איסורי ביאה פט"ו הל' לד) concluded the Imrei Moshe that "He follows his mother and we do not consider the father at all" ("הרי הוא כאמו ואין משגיחין על האב"),
However, halachically, there is a (huge) Machloikes Rishonim (and achronim). Rashi in Yevamos thinks that Ravina there thinks that the child is "Yisrael Gamur" and so thinks Tosfot (on the spot).
BUT - Rashi in Kidushin (p. 72) thinks that Ravina says that he follows his mother to be ruled out as a Mamzer, and therefore the "Kosher Child" din is not met according to Ravina and thus he is considered a Gentile! The same might be understood from Rashi's words: "ויצא בן האשה הישראלית – מלמד שנתגייר" - that implies that he had to go through the Giur process (Note that the Ramban clarifies that the meaning is not an actual Giur).
The same opinion can be found in Tosfot (Kidushin p 71) and Shaar HaMelech (הל' איסורי ביאה פט"ו הל' ט) explaining the Riff's opinion.
And there are many Achronim who ponder on this issue and explain (it's a little too much for this forum)
Some Achronim still fear for the other opinion, some of which are the Maharshal on the Tur (266), and on the Drisha there. Another opinion that fears for it is Rabi Akiva Iger who states that even though he is Israel, we fear for the other opinion. Another one is the Yaarot Devash (Ch 1 last Drush).
According to this, if a Giur Lechumra will not be conducted, some will say the a Bris cannot be done on Shabes, some other will even fear for Lechumra on any gentile\Jewish Halacha...
And some will fear for the Mamzer opinion, which sadly cannot be solved by the Giur Lechumra (hapilly though, as shown above, is not a de-facto fear).
-- If you read Hebrew you can gain insight on many mefarshim, in great details from this wonderful Q&A in the DIN website