Can we terminate Employee who is under probation, with immediate Effect?

Can we terminate an employee who is under probation with immediate effect? There's one employee in my company who has been found very rude, arrogant, misbehaving with teammates, and always giving back answers. The manager of the team has given him verbal warnings twice. Now, the manager has escalated the issue to HR and wants to terminate the employee with immediate effect. Kindly advise whether it is legal or if, as an HR professional, I have to send him a warning letter and then terminate him on his next offense. There has been no email communication prior.

Can you please share the Probation & Termination clause of the appointment letter acknowledged by the said employee.

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DOJ - 14th Nov 2017.
Probation period 1 Year
Termination clause: as per appt letter:
Termination of employment
During the probationary period and any extension thereof, your services may be terminated on either side by giving one month’s notice or salary in lieu thereof. However, on confirmation the services can be terminated from either side by giving one month (30 days) notice or gross salary in lieu thereof.
Notwithstanding the above condition, the contract of service may also be terminated because of under mentioned stipulations. This will be without payment
of any compensation.
(a) If you fail, refuse or neglect to carry out and perform your duties assigned to you by the company. .
(b) For loss of confidence in you by the company for any of the act committed by you.
(c) If you are found to be guilty of fraud, insubordination or misconduct whether in course of performance of duties entrusted to you or otherwise.
(d) If you are found unfit for being entrusted with the responsible work commensurate with your position in consequences of any misconduct, moral turpitude.
(e) If you commit any act prejudicial to the continuing good relationship between you and the company.
(f) If you become physically in capacitated for the performance of your duties.
(g) If you commit breach of any of the terms of this letter of appointment.
Upon termination of employment, you will immediately hand over to the Company all correspondence, specifications, formulae, books, documents, market data, cost data, drawings, affects or records belonging to the Company or relating to its business and shall not retain or make copies of these items.
Upon termination of employment, you will also return all company property, which may be in your possession.

Give him one month's notice as per the appointment order and send him off.

Alternatively, conduct a domestic enquiry and terminate him as he is found to be violating Clauses C and D above. His actions are insubordinate and indisciplined. His team spirit is also lacking. If found guilty after following principles of natural justice, then he can safely be terminated if the company feels it is appropriate. Adequate documentation is needed to avoid any legal hassle subsequently.