I have a point on my scatter plot which seems to be the outlier. I did some calculations and the y-coordinate turned out to be the outlier among the other values of dependent variable, but the x coordinate did not. Can I still call that an outlier and omit it from my analysis?
To be an outlier, the point needs to vary significantly from the normal relationship between the dependent and independent variables. Whilst this may be identified by a scatterplot, they'd need to be confirmed by calculations, which you have performed. Even having only one coordinate off is enough to be an outlier, which is what I see the majority of the time. Here is an example; note that this would need calculations to support it.
Whilst you CAN remove outliers, that doesn't mean you SHOULD delete them. The question of should you removed outliers is answered very well on a different stackexchange thread.
As the (surprisingly deleted) Mattew Borg's answer and it's first picture showed, this point can be an outlier while neither of its coordinates being an outlier.
However, being an outlier is not a reason enough for itself to exclude a point from analysis. In fact, being an outlier is a reason for a point being examined individually to find why an outlier has been produced.
The outlier may be just a measurement error and not to be of interest. For example, this happen when we measure repeatedly the same magnitude, all measures should be around the real value and any departure from it is just because of measurement errors. We can just discard outliers as larger errors.
The outliers may reflect some variation in the underlying process, and discarding outliers would mask those variation. For example, if we are measuring strength of a new type of steel and a few outliers yield a very low value, it might just be showing that there if that type of steel is used, some pieces can be very weak (for example, because of cracks). We can't just discard those outliers, although we may want to perform different analysis for different subpopulations (we can end describing strength of that steel as something like "a normal with exceptions").