Hello !! 

Yes, of course you are eligible for bsc (psychology) but you cannot become a psychiatrist without having biology as a subject . the most you can do with psychology is to become a psychologist. This option is even available in BA also. So you can choose the course according to your interest if you want to go for ba honors in psychology , it is also a good choice 

Good luck!! 

Hey,
B.Sc in Psychology is a three year long undergraduate programme. This course prepares you to investigate the dynamics and inter relations between human beings and the environment including human and machine interface.
The eligibility criteria to pursue Bachelor of Science in
Psychology is to pass 12th from a recognised board with 50 per cent marks in aggregate and have reasonably good problem solving skills and Mathematical skills. So, you are eligible for this course.
The admission is mostly through your 12th marks but some Institutes conduct their own entrance test.
Some of the institutions offering this course are Jamia Milia Islamia, Bananas Hindu University,Jawaharlal Nehru University.
Hope this helps.
Good luck.