Is it okay to do a Brit Shalom instead of Brit milah?

Question

Brit shalom is basically Brit mila without the circumcision, is that okay according to Halacha?

Answer 1

No. Halacha requires Jewish males to be circumcised. The Shulchan Aruch states this with unusual strength in Yoreh Deya' 260-261:

מִצְוַת עֲשֵׂה לָאָב לָמוּל אֶת בְּנוֹ, וּגְדוֹלָה מִצְוָה זוֹ מִשְּׁאָר מִצְוֹת עֲשֵׂה.‏

אִם לֹא מָל הָאָב אֶת בְּנוֹ, חַיָּבִים בֵּית דִּין לְמוּלוֹ. וְאִם לֹא מָלוּהוּ בֵּית דִּין, חַיָּב הוּא, כְּשֶׁיַּגְדִּיל, לָמוּל אֶת עַצְמוֹ. וְאִם לֹא מָל, חַיָּב כָּרֵת.‏

It is a positive commandment for the father to circumcise his son, and this commandment is greater than other positive commandments.

If the father did not circumcise his son, the court is obligated to circumcise him. And if the court did not circumcise him, he is obligated, when he grows up, to circumcise himself. And if he did not circumcise [himself], he incurs karet.

^{My translation.}

For more details about the laws and great significance of Brit Mila, see this writeup by R' Eliezer Melamed. Here are a couple of brief excerpts about this practice's significance:

And just as the first commandment which the patriarch Abraham fulfilled was that of circumcision, so the first commandment that each and every Jewish male who reaches the tender age of eight days old fulfills is that of circumcision. Indeed, this obligation symbolizes, more than any other religious duty, the eternal bond between the Jewish people and their God, a bond which is sealed upon a Jew’s very skin.

...

It is no coincidence that this particular commandment is embellished with great adornment by all Jews, regardless of affiliation to movement and organization. Even if the Jew’s natural bond to some of the commandments has been weakened, when it comes to "Milah" there is a general consensus. This agreement is equivalent to the testimony of a hundred witnesses regarding the true feeling of each Jew regarding Jewish faith and the Torah. ... [T]he precept of "Brit Milah" is undoubtedly the most widely embraced of the commandments, for, more so than any other ritual, it gives expression to a sense of belonging to the Jewish people - the nation which has been chosen for the task of revealing Divine ideals in the world.

Answer 2

Brit shalom is basically Brit mila without the circumcision, is that okay according to Halacha?

Isaac Moses is completely correct that the answer is almost always "no". However, under certain unusual circumstances, it is possible for a male to enter into the covenant of mila (b'ris mila) without undergoing circumcision. This occurs when the person already was circumcised (but not in a way that halachically qualifies as mila) and cannot be circumcised again, or was born without a foreskin. Then the mohel (mila surgeon) will, instead of performing circumcision, do something else that qualifies as mila and enters the person into the covenant: he will draw a drop of blood from the usual site of the foreskin (see that linked-to page for more information and sources).