Am I eligible to join Merchant Navy after Computer Science Engineering???

Yes of course, you can apply for the Merchant Navy after Computer Science Engineering, the minimum qualification requirements for the Merchant Navy examination is that the applicant should have completed 10+2 with PCM (Physics, Chemistry, Mathematics).

You can apply for following training courses:
Diploma in Marine Engineer
Electro technical officer
Graduate Marine engineers
Graduate entry schemes
Nautical science
Deck cadre etc...,

To apply in the merchant navy Candidates have to be unmarried Indian citizens (male or female).