The Shulhan Aruch (Siman 244) prohibits hiring a Non-Jew to do work on Shabbat in a public domain, e.g. constructing a house, even if contracted on a per-project basis. The Shulhan Aruch, follows the Rambam, as well as the Ramban and Ritva, that the reason for the prohibition is Marit Ayin, i.e. other Jews may misconstrue the work as being done in a prohibited fashion, e.g. on a daily basis. Accordingly, if there would not be an issue of outside observers, e.g. if the construction was taking place in a Non-Jewish area, where no Jews lived within walking distance on Shabbat, it would be permitted.
According to this logic, in a place where the prevalent custom is to build houses by contract, and therefore, people are not going to suspect that he hired a day worker, there would be no issue of Marit Ayin, and it would be permitted. This is, in fact, the opinion of the Pri Megadim. The Noda B'Yehuda, Rav Yechezkel Landau in his responsa and the Gaon Rabbi Akiva Eiger brings such an argument, as well.
However, on the other side of this coin, we have the opinion of Rabbeinu Nissim, the Ran. The Ran understands the reason for the prohibition to build a house on Shabbat by contract is not because of Marit Ayin-what other people may think. Rather, the concern is that the owner himself is not going to understand the Halachic difference between Kablanut (project contract) and Sechir Yom (day-worker). He is likely to start mistakenly hiring workers on a daily basis. Accordingly, there would be no room for leniency in a place where the prevalent custom is to build through contract.
The Be’ur Halacha (Rav Yisrael Meir Kagan of Radin, 1839-1933) brings both sides but does not want to give a leniency in such a case, because of other Rishonim that agree with the Ran. The Noda B'Yehuda, although leaning to leniency, concluded to permit only on Hol Hamoed and not on Shabbat.
Rav Moshe Feinstein (Russia-New York, 1895-1986) was very against building on Shabbat. He said the Shabbat is already so weak in the eyes of the people. If you start doing things that the people don't understand, like this, it's only going to weaken the Shabbat more.
On the other hand, Rabbi Moshe Halevi (Israel, 1961-2001), in his Menuhat Ahaba, was lenient. Hacham Ovadia understands the side to be lenient. Halacha l'ma'aseh, one must ask a competent authority before he wants to build on Shabbat with a Non-Jew b'kablanut. It is not something you can just say it's mutar.
SUMMARY
Building a house on Shabbat, where the prevalent custom is to contract the entire project, cannot be permitted without consulting a competent rabbinic authority.
