Possible reasons:
Your validation set is very small compare to your trainning set which usually happens. A little change of weights makes validation loss fluctuate much more than trainning loss. This may not neccessary mean that your model is overfiting. As long as the overall trendency of validation loss keeps decreasing.
May be your train and validation data are from different sources, they may have different distributions. This may happen when your data is time series, and you split your train/validation data by a specific timestamp.
Does the validation loss affect the training process?
No, validation(forward-pass-once) and training(forward-and-backward) are different processes. Hence a single forword pass does not change how would you train next.
Will the algorithm look at the validation loss and slow down the learning rate in case it fluctuates alot?
No, But I guess you can implement your own method to do so. However, one thing should be noted, the model is trying to learn the best solution to your cost function which are fed by trainning data only, so changing this learning rate by observing validation loss doesnt make too much sense.
How can i make the model more stable so that it will return a more stable values of validation loss?
The reasons are expained above. If it is the first case, enlarge validation set will make your loss looks more stable but it does NOT mean it fits better. My suggestion is as long as your are sure your model does not overfit (gap between train loss and validation loss are not too large ), you can just save the model which gives the lowest validation loss.
If its the second case, it can be complecated depend on your case. You could try to exclude samples in trainning set which are not "similar" with your validation set, or enlarge your model's capacity if you have enough data. Or perhapes add more metrics to monitor how well the training.