I want to get spiked gloves as a weapon but my DM says if I have spiked gloves, that counts as a normal weapon and I can't use my unarmed strike with my hands. Is it true?
Yes, your DM is correct, you cannot use unarmed strikes with your hands
There is a decent amount to unpack in this answer, so I've broken it up and addressed each portion directly.
Do spiked gloves count as unarmed strikes?
Unarmed strikes themselves are described on page 195 of the PHB:
Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons). On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your Strength modifier. You are proficient with your unarmed strikes."
As you are wearing your spiked gloves, and your spiked gloves are weapons1 you cannot make an unarmed strike with your hands unless you remove the spiked gloves (as part of your free object interation on your Attack action).
How do spiked gloves interact with Monk class features
I am assuming that you want to make an unarmed strike with your hands because you want it to interact with the Monk Martial Arts feature, but wielding these gloves will cause you problems here.
The text relating to Martial Arts unarmed strikes states (PHB pg 78):
You gain the following benefits while you are unarmed or wielding only monk weapons and you aren't wearing armor or wielding a shield
As mentioned earlier spiked gloves are weapons. As they are not monk weapons you cannot get any of the Monk Martial Arts benefits for any of your unarmed strikes while you are using the spiked gloves.
In particular you cannot activate this Martial Arts feature:
When you usc the Attack action with an unarmed strike or a monk weapon on your turn, you can make one unarmed strike as a bonus action.
What weapons are "monk weapons"?
The is one place in the PHB that defines monk weapons (PHB pg 78):
[...] monk weapons, which are shortswords and any simple melee weapons that don't have the two-handed or heavy property. [...] Certain monasteries use specialized Forms of the monk weapons. For example, you might use a club that is two lengths of wood connected by a short chain (called a nunchaku) or a sickle with a shorter, straighter blade (called a kama). Whatever name you use for a monk weapon. you can use the game statistics provided for the weapon in chapter 5.
Weapons are divided into three categories:
and two sub categories:
Since they have been crafted as a weapon the spiked gloves clearly are not improvised weapons. They are also clearly melee weapons.
So the remaining question to ask is are they simple weapons?
The default list of simple melee weapons is available on page 149 of the PHB. The DM can determine that other weapons are also simple melee weapons, but our default position should be that the list presented in the PHB is exhaustive for the purposes of the game.
Spiked gloves are not on that list (nor could they be reasonably be construed to be a specialised form of one of those weapons).
As a result, unless your DM rules otherwise, they are not monk weapons as they are not simple melee weapons.
Your DM has made a ruling, and as a result, for your game your DMs decision would stand, regardless of what the rules actually say.
1. You stated that you wanted to use spiked gloves as a weapon, so we will assume that they have been made to be weapons in the game you are playing in.
Spiked gloves are not listed in the weapons table (PHB 149) and therefore all their properties are ultimately up to the DM.
However, you may try to persuade him by asking the question: would spiked gloves be a martial weapon? All known weapons are classified as either simple, martial or improvised, and
monk weapons [..] are shortswords and any simple melee weapons that don't have the two-handed or heavy property. (PHB 78)
Spiked gloves are obviously neither two-handed, nor heavy. They were crafted to be used as weapons, so they are not improvised either. About martial weapons:
Martial weapons [..] require more specialized training to use effectively. (PHB 146)
Using spiked gloves is not that much different from striking unarmed and all characters are proficient in unarmed strikes. Why would the gloves "require specialized training"?
But do not forget that while you are free to persuade, do not argue, as the last word is the DM's.
I would say that Spiked Gloves do require special training, and makes them incompatible with Monk's Unarmed Strikes.
Untrained Unarmed Strike do only 1 damage, whereas Monk training upgrades them to at least 1d4, and allows you to use the DEX modifier for bonus attack and damage.
This indicates that a trained Unarmed Strike is more than punching someone. You're doing something that is causing significantly more damage (potentially several times), and allows you to do additional damage via DEX and not just raw STR.