to do some shopping

Vincent TeoCan I say,He walks to the shops to do some shopping.

yes

Can I say,
He walks to the shops to do some shopping.

That is not the correct way to write your question. (And where is the question mark?) You have two ways to ask it correctly.
1) Can I say the following?
He walks to the shops to do some shopping.

2) Can I say 'He walks to the shops to do some shopping.'?
The single quotation marks are

canadian45Can I say 'He walks to the shops to do some shopping.'?

Without the period, of course.

Aspara GusWithout the period, of course.

Why do you say that?

The question mark is the end punctuation of the sentence.

2) Can I say 'He walks to the shops to do some shopping.'?Aspara GusThe question mark is the end punctuation of the sentence.

Yes, and the period is the end punctuation of the sentence being asked about. That period can't be omitted.

canadian45That period can't be omitted.

It can, and it very often is, in my considerable reading experience.

Include the comma and omit the period.

Can I say, "He walks to the shops to do some shopping"?

What if I said to you, "You've got a real problem here"? (Notice that the question mark here comes after the quotation mark and there is no period at the end of the statement.)

DoctorIfInclude the comma

The comma is optional, I’d say. I wouldn’t use it.

As you wish, Asparagus.

It would be more appropriate to include the comma here, that's what I've learnt.