The OP has asked about a father naming his son after himself. This implies that the father is living. Ashkenazim have the custom of not naming a child after any living person. One of the reasons is that there is a halachic problem with a father and son having the same name.
I answer at Naming Children After Living People that this is based on the halachic difficulty in kabeid es Avicha when the father and son have the same name. Note that it is not a halacha not to share the name, but a minhag derived from the halachic problems that could arise. From the name given as an example, it seems that the OP is asking from the Ashkenaz viewpoint. IIRC even Sefardim who do name after a (living) grandfather, do not name children after a (living) father. I will look this up to check for confirmation of my memory.
According to Jewish law it is not deemed proper respect to call one’s
parent by his/her first name.(Yoreh Deah 240:2) Giving a child the
name of the living parent or grandparent would generate confusion and
a belittlement of respect.(Chelkat Yaakov, Yoreh Deah 136,Shmirat
HaGuf V’haNefesh, Vol. II, 154:9)
Originally I only found implicit references to sefardim not naming a child after a (living) father, while they do name after a (living) grandfather.
As an example, Naming Conventions lists the common Sefardi custom of naming as
Common Sephardic naming conventions (also see Jewish Names):
Firstborn son named after the paternal grandfather,
second male child after the maternal grandfather,
first daughter named after the paternal grandmother,
second female child after the maternal grandmother,
next child after the paternal uncle or aunt,
next after maternal uncle/aunt,
Note that the parents are never included in this list as each additional child gets named after a more remote relative. Since this is still only an implication, I will continue to look further.
Thanks to @Ari for finding the reference in the Sefer Chassidim which deals with this question and explains that Jews are makpid not to name a son after the (living father). The Chida - Rabbi Chaim Yosef David Azulai ben Yitzhak Zerachia - on that page says:
And according to all of the minhagim, a person doesn't call his son by the name of himself.
I also found a reference at Names of the Children of Israel by Rabbi Yisrael Kaniel – February 25, 2015 that cites Rabbi Yitzchak Zilberstein (VeHa’arev Na II, pp. 215-218) and R. Menasheh Klein (Teshuvot Mishneh Halachot IX, 248) who discuss this. R. Yehuda Assad (1796-1866) in Yehuda Ya’aleh, Yoreh Deah 247, states that there is a comment by Rashi on Divrei HaYamim I 2:50 that Chur named his son after his own father Kalev who was still alive at the time. In Jerusalem Talmud, Yoma 6:3, we find that Shimon HaTzadik foresaw his own death and instructed his son Nechunya to take his place, thereby sparking the jealousy of his other son Shimon. as well as some other cases.
Rabbi Zilberstein also references the Sefer Chassidim 460 speaks about the Ashkenaz custom while Chida, noted in Brit Olam that,
although some customs allow naming children after living grandparents, all customs agree not to name after oneself. Additionally, R. Assad is found to have addressed the situation of a father named Yitzchak Ber who named his son Ber.
Although it was once accepted to name a child after a living parent, it was no longer the custom, and one must not go against the current prevailing custom.