Actually the opposite.
Its a sign of strength. Have you ever heard the saying "It takes a big man to say he's sorry"?
This is generally stated when the other person realizes just how difficult it was for someone to say their sorry and yet they did.
Even though we have sayings like that where we realize its difficult and shows strength of character to do so it can still be looked upon as a sign of weakness.
It can be seen in some cultures as weakness as it generally involves giving up some strength or position in a group dynamic.
People will think what they will of you regardless. I believe it is better to simply be the best you can be and ignore them. My motto for 50 years has been
Love me, hate me, makes no difference to me.
I will apologize when I feel its necessary, I will live my life according what I believe makes me a good person and that doesn't make me ashamed of myself.
And that is all that is important (to me at least).
If other people my saying sorry or refuse to say it themselves as they perceive it to be a sign of weakness, that is their issue to overcome, not mine.
I will continue saying it when the situation warrants it to be said.